Rationalising the Irrational:
1
How relevant are beliefs for asset prices, credit and
2
aggregate production?
3
Paul De Grauwe and Eddie Gerba∗ December 7, 2017
4
5
We extend the financial accelerator DSGE model to include a powerful feed-back
6
mechanism between asset prices, financial sector and (aggregate) production in an
7
attempt to understand and explain secular stagnation. In particular, we focus on
8
the simultaneous manifestation of anemic production levels, unpaired financial sector
9
and price-to-book ratios below 1 for a sustained time period. Further, we proceed
10
to include the same mechanism in a behavioural macroeconomic New-Keynesian
11
model to contrast and compare the role of beliefs and heterogeneous expectations in
12
amplifying this core machinery and bring it closer to the data. While a key benefit
13
from relaxing rational expectations is a better empirical model fit, there are some
14
important trade-offs in terms of weaker transmission of financial shocks, excessive
15
asymmetry and less transparent and tractable numerical solution method.
16
Keywords: Supply-side, frictions, beliefs, secular stagnation, model validations JEL: B41, C63, C68, E22, E23, E37 ∗ De Grauwe: John Paulson Chair in European Political Economy, London School of Economics, London, WC2A 2AE. Gerba: Research Fellow, London School of Economics, London, WC2A 2AE and Bank of Spain, C/Alcala 48, 28014 Madrid. Corresponding author:
[email protected]. We would like to show our gratitude to the European Commission and the FinMap project for providing the funding. Moreover we would especially like to thank Gianni de Niccolo, Szabolcs Deak, Paul Levine, Cars Homms, and Domenico Delli Gatti for very useful comments and the participating institutions of the European Commission sponsored FINMAP project for their useful insights. Equally, we would like to express our gratitude for comments and remarks received from colleagues at Bank of Spain. The views expressed in this paper are solely ours and should not be interpreted as reflecting the views of Bank of Spain nor the Eurosystem. A previous (but much shorter) version of the paper has been published in the Bank of Spain and CES Ifo Working Paper series.
1
17 18
1
It has long been recognized that the (aggregate) supply-side and financial mar-
2
kets can be powerful generators and propagators of shocks. Yet fewer efforts have
3
so far been invested to understand in what ways and under what conditions the
4
supply side can work as a propagator of shocks generated in the financial sector, or
5
more generally of financial shocks. Nonetheless, a number of recent empirical stud-
6
ies suggest that this interaction is at the core of the contraction in GDP during the
7
Great Recession (see De Grauwe and Gerba (2017) for more details on the empirical
8
findings). Despite the fact that the original negative shocks were generated in the
9
financial sector, a sharp drop in aggregate supply is observed.
10
Meanwhile, a number of Bank of England Financial Stability Reports (2011,
11
2012) and other similar studies on US had shown that productivity growth prospects
12
following the financial market distress in 2008/09 had been so low that it was keeping
13
the stock prices depressed and under its’ fundamental (book) value. On the contrary,
14
a wave of optimism both before that crisis and since 2016 have kept the price-to-
15
book ratios well above 1, leading many to conclude that the stock prices were and
16
are ‘overvalued’. Similar boom-bust cycle is also observed for corporate dividends
17
and investment. In turn beliefs and sentiments play a large role for the real economy
18
as they influence firms’ investment capacity, (bank) financing available to firms, and
19
productivity growth prospects.
20
In standard macroeconomic models, expectations work as powerful amplifiers of
21
cyclical movements by bringing future (state variable) realizations forward, similar to
22
an intertemporal catalyser. So, for instance, in Gerba (2017) rational expectations
23
of investors on the stock market bring future corporate production-and external
24
financing capacity forward by pushing market prices of those corporate stocks up
25
(or down) already today. The final outcome is that fluctuations in the business cycle
26
and the balance sheet variables are intensified by a factor of three compared to a
27
standard financial accelerator model. Yet, despite this increase in swings, second
28
moment analysis shows that the volatilities of many macro-financial variables in the
29
model fall below those observed in the long-run US data.
30
In parallel, a series of other papers have studied the impact of imperfect stock
31
market beliefs on business- and financial cycle swings, as well as the (in)stability of 2
the economy. Amongst many, the model of De Grauwe and Gerba (2017) examines
1
the impact of subjective and heterogeneous expectations of stock market investors
2
on aggregate production, firm financial constraints, and the aggregate cycles. The
3
agents in this behavioural model are intrinsically rational insofar that they dynam-
4
ically optimize and objectively learn from the past, but optimize under subjective
5
beliefs about the aggregate state which they do not control. Compared to a version
6
where there is no link between imperfect (stock market) beliefs and production, the
7
swings are intensified by a factor of two to three, while volatilities are for many
8
variables higher compared to long-run US data.
9
Hence, one would be tempted to conclude that introducing imperfect beliefs
10
in supply-side financial frictions framework results in huge amplification of cycles,
11
drastic increases in volatility beyond those observed in the data, and increased
12
(macro)economic instability. As a result, beliefs generate excessive volatility, aym-
13
metry and tail events not supported by the empirics. Yet, as far as we are aware,
14
a cross-model-consistent and systematic comparison of the impact of beliefs and
15
stock markets on supply side financial frictions has not been performed. In partic-
16
ular, there is a gap regarding the empirical validation of the role that subjective
17
beliefs play for aggregate fluctuations and system-wide stability in the context of
18
macroeconomic models. Our aim in this paper is to fill this gap and provide some
19
insight on the role of supply-side financial frictions in generating deep and sustained
20
contractions, and intense credit (financial) cycles.1
21
1
22
Model set-up
The main theoretical contribution of this paper is to quantify the full impact of
23
a feed-back mechanism between asset prices, financial sector, and the (aggregate)
24
supply side under different information sets. Previous papers have only partially
25
1
A related literature to this one is Justiniano et al (2010, 2011) who investigate the business cycle role of shocks to marginal efficiency of investments, or Altig et al (2011) who explore the importance of firm-specific capital for driving the business cycle. Lastly Smets and Wouters (2005, 2007) Gerali et al (2010) and Christiano et al (2010, 2013) also include a more elaborate supply side in their models, but do not specifically focus on the role of the supply-side as a propagator of financial shocks by altering the input-output ratio, labour productivity, or aggregate output per employee.
3
1
examined this mechanism within an incomplete framework.2 Capitalizing on these
2
models and the insights on the important role of aggregate supply in propagating
3
financial (and other) shocks, we extend the standard financial accelerator model in
4
multiple dimensions. First, we include a stock market with an endogenous price
5
(and return) evolution. Second, we incorporate frictions on the production side of
6
the economy. Third, and final, we introduce information frictions and heterogeneous
7
expectations in one version of the model (meanwhile we keep rational expectations
8
in the other), in order to investigate the role that imperfect beliefs and learning play
9
for the strength and amplitude of the supply side financial frictions. The complete
10
mechanism therefore contains a chain of three reinforcing frictions in three sectors.
11
Figure 1 summarizes the various components of the mechanism introduced in this
12
paper. Information frictions Asset prices on stock market
Aggregate production
Financial sector Financial frictions
Supply side frictions
Figure 1: Fully-fledged mechanism 13
Asset prices in this model represent future return of corporate shares. Investors
14
determine, with their available information, what the output, productivity and divi-
15
dend level of firms will be in the future. The better the investor prospects, the higher
16
the price (and vice versa). The only difference between the two model versions is
17
that in the incomplete information setting, investors estimate these core variables
18
using limited information sets, but rely on multiple rules to forecast them. This
19
gives rise to swings in market sentiment. In rational expectations, this mechanism
20
is absent.
21
In the financial sector, due to information asymmetry-and state verification prob2
See Gerba (2016, 2017) for a detailed discussion on the literature and recent contributions.
4
lems of the financial accelerator type, banks require collateral from borrowers and
1
write state-contingent contracts. This mechanism is introduced identically in both
2
models.
3
On the supply side end, we introduce a production-and financing friction by in-
4
corporating a utilization rate of capital in production, by making a firm’s purchasing
5
position on the input markets conditional on its solvency position. Compared to the
6
standard workhorse financial accelerator model, this includes three extensions. The
7
first extension allows the firm to employ capital in a more intelligent manner as
8
it does not only decide on the quantity of capital to purchase, but also the rate at
9
which it uses (or depreciates) it in production.3 The second extension introduces
10
a pay-in-advance constraint for capital purchases on the input market. We motivate
11
it as a depository insurance scheme for capital good producers since firms need to
12
pay a share of the capital cost in advance of purchase. Firms will finance it with a
13
share of the (liquid) external financing that they get. Since this in turn depends on
14
the cash position that they will hold in the next period, the expected (stock) market
15
price will de facto reflect the price they have to pay in advance for the capital. We
16
make the down payment time varying over the business cycle in order to capture the
17
asymmetries in financial (or liquidity) positions over the cycle. The final extension
18
links (financial) solvency to productivity. A higher value of net worth means that
19
the collateral constraint the producing firm faces is lower. As a result it can borrow
20
more, which will press the marginal costs down, and therefore they will be able to
21
buy capital inputs at a lower price.
22
The intuition tells us that this fully-fledged machinery is significantly more pow-
23
erful in propagating shocks and generates larger cyclical swings than any of the
24
model versions where only parts of this feed-back loop are captured, or some as-
25
pects are introduced exogenously. In general, we should expect that the full device
26
should increase the propagation of shocks by a factoir of two to three, at the same
27
time as it should be able to discriminate between small and large swings, where the
28
latter are three to five times larger.
29
3
The higher the rate, the more effective use is made of capital in the production function and the more (intermediate) products can be produced ceteris paribus. However, increasing the capital utilization cost is also costly because it causes a faster rate of capital depreciation.
5
1
Turning to information frictions, there are of course multiple ways in which their
2
‘added value’ can be tested. One way is to directly introduce (partial) information
3
frictions in the rational expectations DSGE model. We say partial, since by model
4
construction, the DSGE model can only accommodate a specific type of subjective
5
beliefs and heterogeneous expectations. Our aim instead is to test the full amplifica-
6
tion and propagation power of information frictions. For that purpose, we test two
7
extreme versions: one where rational expectations apply to all parts of the model
8
including the stock market, and another where imperfect information and hetero-
9
geneous expectations are introduced on the aggregate states and the key variables
10
that are affected by frictions. Using this as the line of bifurcation, we will call the
11
first a DSGE model, and the second a behavioural.
12
We proceed by incorporating these mechanisms in a DSGE model, followed by
13
the behavioural.
14
1.1
15
The model we use here is mainly based on the framework described in Gerba (2017).
16
It is an (endogenous) asset price extension of the financial accelerator model. In
17
addition to this, we disentangle capital production from capital utilization rate,
18
and introduce variable capital usage.4 Therefore, the production side of the economy
19
consists of three types of nonfinancial firms: capital good producers, entrepreneurs,
20
and retailers. For the sake of brevity, we will only describe these novelties in the
21
main text. For the full system of equations, we refer to the Appendix.
22
1.1.1
23
Following Gerali et al (2010), perfectly competitive capital good producers (CGP)
24
produce a homogeneous good called ’capital services’ using input of the final output
25
from entrepreneurs (1 − δ)kt−1 and retailers (it ) and the production is subject to
26
investment adjustment costs. They sell new capital to entrepreneurs at price Qt .
27
Given that households own the capital producers, the objective of a CGP is to choose
28
Kt and It to solve: 4
The DSGE model
Capital Good Producers
For the remaining model set-up, we refer to aforementioned paper.
6
max E0 Σ∞ t=0 Λ0,t [Qt [Kt − (1 − δ)Kt−1 ] − It ] Kt ,It
(1)
subject to:
1
κ It Kt = (1 − δ)Kt−1 + [1 − [ − 1]2 ]It 2 It−1
(2)
it where [1 − κ2 ( it−1 − 1)2 ]It is the adjustment cost function. κ denotes the cost for
2
adjusting investment. Including adjustment costs of investment in the production
3
of capital solves the so-called ’investment puzzle’ and produces the hump-shaped
4
investment in response to a monetary policy shock (Christiano et al, 2011).
5
1.1.2
6
Entrepreneurs
Perfectly competitive entrepreneurs produce intermediate goods using the constant
7
returns to scale technology:
8
Yt = At [ψ(ut )Kt ]α L1−α
(3)
with At being stochastic total factor productivity, ut the capacity utilization rate,
9
and Kt and Lt capital and labor inputs. Capital is homogeneous in this model.5 We
10
assume a fixed survival rate of entrepreneurs in each period γ, in order to ensure
11
a constant amount of exit and entry of firms in the model. This assumption also
12
assures that firms will always depend on external finances for their capital purchases,
13
and so will never become financially self-sufficient.
14
Just as in the canonical financial accelerator model (Bernanke, Gertler and
15
Gilchrist, 1999) as well as in the extension (Gerba, 2014), we will continue to assume
16
that all earnings (after paying the input costs) from production are re-invested into
17
the company such that a constant share is paid out to shareholders.6 This is why
18
entrepreneurs will maximize their value function rather than their production func-
19
tion.7
20
5
We could have made capital firm-specific, but the set-up would have to be much more complex without altering qualitatively the results. Using homogeneous capital assumption is standard in these type of models, see for instance Bernanke et al (1999), Gerali et al (2010), Gertler et al (2012). 6 In our exercises, we will set this share to 0, just as in Bernanke et al (1999). 7 And so yt is not a direct argument of the function.
7
1
Entrepreneurs also choose the level of capacity utilization, ψ(ut ) (Kydland and
2
Prescott (1988), Bills and Cho (1994)). As is standard in the capital utilization liter-
3
ature, the model assumes that using capital more intensively raises the rate at which
4
it depreciates.8 The increasing, convex function ψ(ut )kt denotes the (relative) cost in
5
units of investment good of setting the utilization rate to ut . This is chosen before
6
the realization of the production shock (see Auernheimer and Trupkin (2014) for
7
similar assumption). This timing assumption is important because it separates the
8
choice of the stock of productive factor Kt , taken before the revelation of the states
9
of nature, from the choice of the flow of factor ut Kt , taken during the production
10
process.
11
The choice of the rate of capital utilization involves the following trade-off. On
12
the one hand, a higher ut implies a higher output. On the other hand, there is
13
a cost from a higher depreciation of the capital stock. Therefore this rate can be
14
understood as an index that shows how much of the stock of capital is operated
15
relative to the steady state, per unit of time, given a capital-labor services ratio.
16
Moreover we specify the following functional form for ψ(ut ):
ψ(ut ) = ξ0 + ξ1 (ut − 1) +
ξ2 (ut − 1)2 2
(4)
17
in line with Schmitt-Grohe and Uribe (2006), Gerali et al (2010), and Auern-
18
heimer and Truphin (2014). As a result, an entrepreneur will now maximize his
19
value (profit) function according to:
V = max E0
∞ X k=0
Z [(1 − µ)
$ rk ks ωdF ωUt+1 ]Et (Rt+1 )St ψ(ut )Kt+1 − Rt+1 [St Kt+1 − Nt+1 ]
0
(5) 20
with µ representing the proportion of the realized gross payoff to entrepreneurs’
21
capital going to monitoring, ω is an idiosyncratic disturbance to entrepreneurs’
22
ks return (and $ is hence the threshold value of the shock), Et Rt+1 is the expected
23
rk stochastic return to stocks, and Ut+1 is the ratio of the realized returns to stocks to 8
We could equally assume a fixed rate of capital depreciation and impose a cost in terms of output of using capital more intensively, as in Christiano et al (2005) or Gerali et al (2010).
8
ks ks ]). /Et [Rt+1 the expected return (≡ Rt+1
1
To understand how a firm’s financial position influences its’ purchasing power in
2
the capital input market, we need to understand the costs it faces. A firm minimizes
3
the following cost function:
4
s S(Y ) = min[Rt+1 Kt + w t L t ]
(6)
k,l
The real marginal cost is therefore s(Y ) =
s(Y ) =
∂(S(Y ) , ∂(Y )
which is:
1 1−α 1 α s α (r ) (wt )1−α 1−α α t+1
s = The return on capital is defined as Rt+1
5
(7)
Et [St+1 ]−St 9 . Keeping St
the wage rate
6
constant, an increase in the expected (stock) market value of capital reduces the
7
(relative) cost of capital service inputs, purchased at today’s capital price.10 This is
8
easier to see in the entrepreneur’s budget constraint:11
9
ϑEt [St+1 ]Kt+1 + wt Lt + ψ(ut )kt−1 + Rt Bt−1 + (1 − ϑ)St Kt =
Yt + Bt + St (1 − δ)Kt−1 ⇒ Xt
ϑEt [St+1 ]Kt+1 + wt Lt + ψ(ut )Kt + Rt [St Kt − Nt ] + (1 − ϑ)St Kt = Yt + [Et [St+1 ]Kt+1 − Nt+1 ] + Et [St ](1 − δ)Kt−1 (8) Xt with δ being the depreciation rate of capital, ψ(ut )Kt−1 the cost of setting a level 9
Following on from Gerba (2014) and disentangeling Tobin’s Q, we define the (stock) market value of capital St as the total value of the firm, including intangibles, meanwhile the book value Qt is the accounting value of the firm that includes tangibles only. The difference between the two is the residual earnings REt , which varies positively with (expected) firm performance and economic prospects. Market value of capital determines the level of firm (physical) investments. As a result, periods of high price-to-book ratios and positive economic outlook will drive investment up by significantly more than in standard DSGE models. In return, when corporate and economic outlook worsen, investor confidence on the stock markets will fall, driving down the market value of capital, and therefore also (physical) investment. For a more detailed background on capital prices and a discussion of macroeconomic implications of this stock market mechanism, refer to Gerba (2014). 10 In line with the costs that intermediate firms face in the model of Christiano et al (2005). 11 We assume that entrepreneurs borrow up to a maximum permitted by the borrowing constraint.
9
10
Ptw Pt
=
1 Xt
is the
1
ut of the utilization rate, ϑ is the front payment share to CGP, and
2
relative competitive price of the wholesale good in relation to the retail good.12 An
3
increase in the expected market price (right-hand side) has two effects. First, it
4
reduces the relative cost of capital purchases today since firms can borrow more and
5
pay a higher pre-payment share ϑ of capital. Second, a higher market price means
6
that the probability of default of an entrepreneur reduces (since the value of the firm
7
is higher) and so CGP will expect entrepreneurs to be solvent in the next period
8
and will therefore require a smaller front payment (i.e. ϑ on the left-hand side will
9
fall). Let us explain the second mechanism in further detail.
10
As a form of depository insurance, CGP will (in some periods) require en-
11
trepreneurs to pay in period ’t’ a share of the total capital produced and delivered to
12
entrepreneurs in period ’t+1’. In particular, when CGP suspect that entrepreneurs
13
will face liquidity problems in the next period, a lower production, or a lower col-
14
lateral value in the next period, they expect the firm to be less solvent (in relative
15
terms). Because the default probability of entrepreneurs rises, CGP become suspi-
16
cious of the entrepreneur’s ability to pay for the entire capital purchased. Therefore,
17
as an insurance mechanism, CGP will ask the entrepreneur to pay in advance a share
18
of its capital production.13
19
If the entrepreneur’s value is expected to increase in the next period, the financing
20
constraint he faces will loosen, and thus he can borrow more. Since he can borrow
21
more, he has more money to purchase the inputs (i.e. the marginal cost of a unit
22
of capital decreases, ceteris paribus) and therefore produce more outputs. This will
23
push the price of capital in the future up. The CGP anticipating this, will require
24
a smaller share of capital production to be pre-paid. On the other end, if the value
25
of the firm is expected to fall, on the other hand, then the cost of financing will
26
increase and the firm will be able to borrow less. Because it can borrow less , it
27
has less money to purchase inputs, and this will push the price of capital down in
28
the future. In anticipation of this, CGP will require a higher front payment. Hence, 12
Note that ϑEt [St+1 ]Kt+1 < Et [St+1 ]Kt+1 . We could equivalently assume that legal conditions/constraints stipulate that entrepreneurs need to pay in advance for their inputs as in Champ and Freedman (1990, 1994). Our approach is analogue to the one taken in Fuerst (1995) or Christiano and Eichenbaum (1992) for labor input costs. 13
10
we expect the share ϑ to vary over the business cycle. Formally, the pay-in-advance
1
constraint that entrepreneurs face in the input market is:
2
Et [St+1 ]Kt+1 ≤ ϑt Bt ≡ ϑt
Et [St+1 ]Kt+1 Nt
(9)
So the down payment share of capital purchases will depend on the entrepreneur’s
3
financial position Bt . We can equivalently express it in terms of the additional
4
external funds that the entrepreneur needs for its capital purchases (right-hand side
5
in the above expression) using the fact that an entrepreneur will borrow up to a
6
maximum and use it to purchase capital:14 We allow ϑ to vary over time in order to
7
capture the variations in CGP’s pre-cautionary motive over the business cycle. A
8
value of 1 means that the entrepreneur will need to use all of his external finances
9
(loan) to pay for the capital purchases since CGP expects its financial (cash) position
10
to worsen in the next period. Equivalently, a value of 0 means that no pre-payment
11
is required as CGP expects the entrepreneur to be able to pay in full for its purchases
12
in the next period. As a result, the constraint will not be binding.
13
Both the individual and aggregate capital stock evolves according to:
Kt = (1 − δψ(ut ))Kt−1 + Ψ(
It )Kt−1 Kt
14
(10)
where Ψ( KItt )Kt−1 are the capital adjustment costs in the usage of capital. Ψ(.)
15
is increasing and convex, and Ψ(0) = 0. The term δψ(ut ) follows Burnside and
16
Eichenbaum (1996) and represents the endogenous capital depreciation rate, which
17
is important for the propagation of productivity shocks (see Greenwood et al (2000),
18
or Albonico et al (2014)).15
19
Besides production, capital purchase, cost minimization and credit demand, en-
20
trepreneurial expected production capacity and entrepreneurial net worth is evalu-
21
ated by the (rational) stock market investors. Following precisely the mechanism
22
described in Gerba (2016), the stock market provides (anticipated) liquidity to en-
23
14
See Bernanke et al (1999) and Gerba (2014) for a more profound discussion of the entrepreneur’s capital demand behaviour. 15 The log-linearized version of this expression is: kt = (1 − δψ(ut ))kt−1 + δit , as in Bernanke, Gertler and Gilchrist (1999) or Gerba (2014) and the one used in the simulations. δit is the steady It )Kt−1 . state version of Ψ( K t
11
1
trepreneurs if he believes that the entrepreneur’s expected production and cash
2
growth is positive. This in turn pushes net worth of entrepreneur up, which allows
3
him to borrow more on the credit market, and so on. In addition, entrepreneurs
4
supply labour on the goods market. However, following the logic of the canonical
5
financial accelerator model, the share of labour income to total entrepreneurial net
6
worth is calibrated to a minimum The reminder of the model is the same as in
7
Bernanke et al (1999) and Gerba (2016). In other words, households use their wage
8
income to consume and save. They charge financial intermediaries a cost on their
9
deposits both to cover for the opportunity cost of depositing the money and the
10
cost for the possibility of financial intermediaries investing money in less produc-
11
tive projects. Financial intermediaries, in turn, use deposits to give out as credit
12
to entrepreneurs in a perfectly competitive credit market. However, they charge
13
entrepreneurs a finance premium for the required monitoring of this credit and for a
14
(possible) credit default. To complete the model, retailers buy intermediary goods
15
from entrepreneurs, transform them and sell them as final goods to consumers on
16
a monopolistically competitive goods market a la Calvo. Lastly, the government
17
produces money, runs fiscal stimulus and collects taxes, but does not run deficits.16
18
1.2
19
Our next task is to incorporate the same mechanism in the behavioural model.
20
This has broadly been done in De Grauwe and Gerba (2017) so we use their model
21
as our workhorse behavioural tool. We even use the same asset price evolution
22
as in their paper. With respect to the DSGE model before, the only difference is
23
that agents have cognitive limitations in this framework and thus optimize using
24
imperfect knowledge regarding some aggregate states. However, they use multiple
25
simple heuristics rules (rules of thumb) to approximate these states, and objectively
26
evaluate them. Therefore, they learn and improve their forecasting performance
27
over time. As will be shown later, this switching in rules and learning plays an
28
important role in generating waves in market sentiments and diverse boom-bust
The behavioural model
16
This final section of the model is exactly the same as in the original Bernanke et al (1999) model.
12
cycles.17 In what follows, we will briefly outline the full feed-back mechanism in this
1
paper, largely in line with De Grauwe and Gerba (2017).
2
Capital good producers (CGP) produce capital which they rent to entrepreneurs
3
at a cost Rts . Entrepreneurs use the newly purchased capital and labor to produce
4
final goods. Whereas capital good producers and entrepreneurs operate in perfectly
5
competitive goods markets, retailers are monopolistically competitive. Therefore
6
they price discriminate, resulting in price frictions on the aggregate supply side
7
(Phillips curve). To avoid repetition, we will only briefly describe the optimization
8
problem of entrepreneurs since most of it is identical to entrepreneurs in the DSGE
9
model. For CGP, we refer the reader to the description in the DSGE section since
10
they are identical in both model versions.
11
1.2.1
12
Entrepreneurs
Perfectly competitive entrepreneurs produce intermediate goods using the constant
13
returns to scale technology:
14
Yt = At [ψ(ut )Kt ]α L1−α
(11)
with At being stochastic total factor productivity, ut the capacity utilization rate,
15
and Kt and Lt capital and labor inputs. Capital is homogeneous in this model. We
16
assume a fixed survival rate of entrepreneurs in each period γ, in order to ensure
17
a constant amount of exit and entry of firms in the model. This assumption also
18
assures that firms will always depend on external finances for their capital purchases,
19
so they will never become financially self-sufficient.
20
Entrepreneurs also choose the level of capacity utilization, ψ(ut ). The increas-
21
ing convex function ψ(ut )kt denotes the (relative) cost in units of investment good
22
of setting the utilization rate to ut . This is chosen before the realization of the
23
production shock (see Auernheimer and Trupkin (2014) for similar assumption).
24
Moreover we specify the following functional form for ψ(ut ): 17
One of the core questions will of course be whether it is necessary to turn to strict bounded rationality in order to create realistic non-linearities in the transmission of shocks and asymmetric business cycle fluctuations. The empirical fit is the next step in the comparison.
13
25
ψ(ut ) = ξ0 + ξ1 (ut − 1) + 1
2
ξ2 (ut − 1)2 2
(12)
in line with Schmitt-Grohe and Uribe (2006), Gerali et al (2010), and Auernheimer and Truphin (2014).18
3
To understand how a firm’s financial position influences its’ purchasing power in
4
the capital input market, we need to understand the costs it faces. A firm minimizes
5
the following cost function:
S(Yt ) = min[Rts Kt + wt Lt ]
(13)
k,l
6
The real marginal cost is therefore s(Yt ) =
∂S(Yt ) , ∂(Yt )
which is:
1 1−α 1 α s α s(Yt ) = (rt ) (wt )1−α 1−α α 7
The gross return on capital is defined as Rts =
St −St−1 . St−1
(14) Keeping the wage rate
8
constant, an increase in the (stock) market value of capital reduces the (relative)
9
cost of capital service inputs, purchased at today’s capital price.19
10
This is easier to see in the entrepreneur’s budget constraint:20
St+1 Kt+1 + wt Lt + ψ(ut )kt−1 + Rt Bt−1 + (1 − ϑ)St Kt =
Yt + Bt + St (1 − δ)Kt−1 ⇒ Xt
St+1 Kt+1 + wt Lt + ψ(ut )Kt + Rt [St Kt − Nt ] + (1 − ϑ)St Kt = Yt + St+1 Kt+1 − Nt+1 ] + St (1 − δ)Kt−1 (15) Xt 11
,where:
Bt = St Kt − Nt 18
(16)
In the simulations, ut will be normalized such that ψ(ut ) = 0.80. In line with the costs that intermediate firms face in the model of Christiano et al (2005). 20 We assume that entrepreneurs borrow up to a maximum permitted by the borrowing constraint. 19
14
or Bt is the amount entrepreneurs borrow, δ being the depreciation rate of cap-
1
ital, ψ(ut )Kt−1 the cost of setting a level ut of the utilization rate, ϑ is the share of
2
capital purchases required to be paid in advance by CGP, and
Ptf Pt
=
1 Xt
is the relative
3
competitive price of the final good in relation to the capital good (i.e. mark-up).21 An
4
increase in the (stock) market price (right-hand side) has two effects.22 First, it re-
5
duces the contemporaneous (relative) cost of capital purchases since firms can bor-
6
row more and pay a higher pre-payment share ϑ of capital. Second, a higher market
7
price means that the probability of default of an entrepreneur drops (since the value
8
of the firm is higher) and so CGP will expect entrepreneurs to be solvent in the next
9
period and will therefore require a smaller pre-payment (i.e. ϑ on the left-hand side
10
will fall). Let us elaborate on this second mechanism a bit further.
11
As a form of depository insurance, CGP will in some periods require entrepreneurs
12
to pay in period t a share of the total capital produced and delivered to entrepreneurs
13
in period ’t+1. The share to be paid is strongly contingent on the amount that the
14
entrepreneur can borrow on the credit market Bt .
15
Formally, the pay-in-advance constraint that entrepreneurs face in the input market is:
16
17
St+1 Kt+1 ≤ ϑt Bt ≡ ϑt [St+1 Kt+1 − Nt ]
(17)
We allow ϑ to vary over time in order to capture the variations in CGP’s pre-
18
cautionary motive over the business cycle. In theory, a value of 1 means that the
19
entrepreneur will need to use all of his external finances (loans) to pay for the capital
20
purchases since CGP expects entrepreneur’s financial (cash) position to worsen in
21
the next period. Equivalently, a value of 0 means that no pre-payment is required
22
as CGP expects the entrepreneur to be able to re-pay in full for its purchases in the
23
next period. As a result, the constraint will not be binding. Considering the entire
24
parameter spectrum of ϑ, the constraint will almost always be binding, except for
25
the extreme case when ϑ = 0, or no pre-payment is required at all due to very sound
26
financial conditions of the entrepreneur. In our long-run simulations, the constraint
27
21 22
Note that ϑSt+1 Kt+1 ≤ St+1 Kt+1 . For an explanation of the evolution of stock prices, we refer to Annex I.
15
1
is mostly binding since slightly more than half of the time ϑ is above 0. This implies
2
that in most cases CGP wish entrepreneurs to at least pay a share of their input
3
purchases in advance.
4
Both the individual and aggregate capital stock evolves according to:
Kt = (1 − δ)Kt−1 + Ψ( 5
It )Kt−1 Kt
(18)
where Ψ( KItt )Kt−1 are the capital adjustment costs in the usage of capital. Ψ(.)
6
is increasing and convex, and Ψ(0) = 0.23
7
1.3
8
Next, we need to adapt aggregate state equations to accommodate the production
9
economy. First we need to link capital to real interest rate. Linking the investment
10
Aggregate dynamics
demand equation from De Grauwe and Macchiarelli (2015): it = i(ρ)t = e1 E˜t yt+1 + e2 (ρ − E˜t πt+1 ); e2 < 0
11
12
(19)
with the aggregate capital accumulation 18, we find that the relation between capital and the real rate is:
kt = (1−δ)kt−1 +Ψ(
it it−1
)i(ρ)t = (1−δ)kt−1 +Ψ(
it it−1
)e1 E˜t yt+1 +e2 (rt +xt −E˜t πt+1 ); e2 < 0 (20)
13
14
Incorporating a supply side into the aggregate system of De Grauwe and Macchiarelli (2015) - by combining equations 19, 20 and 12 - gives:
yt = a1 E˜t yt+1 +(1−a1 )yt−1 +a2 (rt −E˜t πt+1 )+(a2 +a3 )xt +(a1 −a2 )ψ(ut )kt +Adjt +t ; (a1 −a2 ) > 0 (21) 15
Note that the first four terms are equal to the aggregate demand expression 23
The log-linearized version of this expression is: kt = (1 − δ)kt−1 + δit , as in Bernanke, Gertler and Gilchrist (1999) or Gerba (2014) and the one used in the simulations. δit is the steady state It version of Ψ( K )Kt−1 . t
16
in De Grauwe and Macchiarelli (2015). Besides that, aggregate demand now also
1
depends on the usable capital in the production, ut kt but discounted for the cost of
2
financing (xt ). Christiano et al. (2005), Smets and Wouters (2007), and Gerali et
3
al. (2010) arrive at the same resource constraint expression in their models. There
4
is an adjustment cost in investment given by equation 20, which we capture by
5
it ) ≡ Ψ( KItt ).24 However, it will be calibrated in such a way to equal δ, Adjt = Ψ( it−1
6
as in standard DSGE models.
7
Monetary policy affects aggregate demand in two ways. First, it affects the op-
8
portunity cost of current consumption and investment, via the third term in equation
9
I.34. Second, it affects return on investment in expression 20, which affects capital
10
demand, captured by the fifth term in the aggregate demand expression. To exam-
11
ine the impact of changes in interest rate on the economy, we introduce a (negative)
12
monetary policy shock ( calibrated to 0.5) with the following autoregressive process:
13
rt = rt−1 + γπt + (1 − γ)yt +
(22)
We will examine the system’s response to this shock in section ??.
14
The reader will notice that aggregate demand also depends on the external fi-
15
nance (or risk) premium xt . This is a reduced form expression for investment, since
16
investment is governed directly by this premium, and therefore it is the dependent
17
variable (see De Grauwe and Gerba (2017) for a derivation of this term).
18
The aggregate supply (AS) equation is obtained from the price discrimination problem of retailers (monopolistically competitive): πt = b1 E˜t πt+1 + (1 − b1 )πt−1 + b2 yt + νt
19
20
(23)
As explained in DeGrauwe and Macchiarelli (2015), b1 = 1 corresponds to the
21
New-Keynesian version of AS with Calvo-pricing (Woodford (2003), Branch and
22
McGaugh (2009)). Setting 0 < b1 < 1 we incorporate some price inertia in the vein
23
of Gali and Gertler (1999). Equally, the parameter b2 varies between 0 and ∞ and
24
reflects the degree of price rigidities in the context of a Calvo pricing assumption
25
24
These two are equivalent since, via the Cobb-Douglas production function, capital has a constant share in production over time and is homogeneous.
17
1
(DeGrauwe, 2012). A value of b2 = 0 corresponds to complete price rigidity and
2
b2 = ∞ to perfect price flexibility (firms have a probability of 1 of changing prices
3
in period t).
4
Agents form expectations regarding output and inflation following the mecha-
5
nism described in the appendix, De Grauwe and Gerba (2017), or De Grauwe and
6
Macchiarelli (2015). Briefly, agents form expectations using two competing rules.
7
The probabilities of choosing one of these rules is determined by the relative fore-
8
casting performance of these rules using the mean squared forecast error (MSFE)
9
criterion.
10
1.4
Forcing variables
11
We will examine the impulse responses to two supply shocks (TFP and utilization
12
rate), one financial shocks (firm financing costs), and one monetary shock (monetary
13
policy). For the DSGE model, we will additionally consider a shock to the asset price
14
wedge. Note that unlike in the DSGE version, the shock processes are not persistent
15
in the behavioural model. Therefore we only introduce a standard white noise shock,
16
without an autoregressive parameter. The persistence in that model comes from
17
the endogenous device itself. Said that, we could easily extend this to include an
18
autoregressive process for the behavioural model as well (even if that is not common
19
nor accepted in this literature), with the consequence of the transmission being even
20
more forceful. Further details on the structure of the five shocks is described in the
21
appendix.
22
2
23
The quantitative analysis is split into three parts. First, we will compare the model-
24
derived statistical moments to the US data. Next, we will examine and quantify the
25
drivers of the business-and financial cycles. For the DSGE model, we will perform
26
a variance decomposition of the simulated paths of the variables conditional on
27
all shocks being activated. The size and persistence of all shocks is kept equal
28
which implies that the decomposition numbers in effect inform on the importance
Quantitative results
18
of each shock in structurally driving the variables. A very similar exercise will also
1
be performed for the behavioural model. However, since cycles are endogenously
2
generated in that model (without the need to insert exogenous shocks), instead we
3
will focus our attention on the structural role of beliefs and learning for driving the
4
variables (and their distributions), both in the time-series and frequency domain
5
after simulating the cycles for 2000 periods (or 500 years). To finish off, we will
6
analyse a selected number of impulse responses, in particular to technology and
7
financial shocks. To maintain the focus of the paper, we have deferred the discussion
8
on the calibration of both models to appendix I of this paper. Moreover, impulse
9
response analysis of the other three shocks (shock to utilization rate and monetary
10
policy in both models, + shock to the asset price wedge in the DSGE model) are
11
available upon request from the authors.
12
2.1
13
Statistical moments in the DSGE model
The statistical moments are reported in Tables 2 and 3. For that, we have calculated
14
the statistical moments for all US variables using the longest data sample period
15
available from 1953:I - 2014:IV.25 Following Stock and Watson (1998), we choose
16
1953:I as the starting year of our sample since the (post-war) quarters prior to
17
1953 include noise and inaccuracies in the data recording. The sample includes 247
18
quarters (or 62 years) which is the closest approximation available for the long-run
19
(cyclical) moments that is generated by the model (see Table 3). During this period,
20
the US economy experienced 10 cycles (using NBER business cycle dates), and the
21
average GDP increase (quarter-on-quarter) during expansions was 1.05% while it
22
was -0.036% during recessions. The data were downloaded from Flow of Funds
23
at the Fed St Louis database. These were de-trended using a standard two-sided
24
HP-filter with λ = 1600 before their moments were calculated.26
25
25
The most recent data recorded is for 2014:IV using Fed St Louis database on March 2, 2015. This is in order to allow for a smoother comparison with the model generated (cyclical) moments. 26
19
Table 1: US variables and sources Variable yt ct it kt ht dt st lts rt xt /ef pt nbt nft πt
US data name Frequency Source Period Real GDP Quarterly Fed St Louis database 1953:I-2014:IV Real Personal Consumption Expenditure Quarterly Fed St Louis database 1953:I-2014:IV Real Investment Quarterly Fed St Louis database 1953:I-2014:IV Capital Stock Annually Fed St Louis database 1953:I-2011:I Hours of All Persons: Nonfarm Business Sector Quarterly Fed St Louis database 1953:I-2011:IV Total Savings and Time Deposits of Households Quarterly Fed St Louis database 1953:I-2014:IV Net Private Savings Households Quarterly Fed St Louis database 1953:I-2014:IV Credit Market Instruments for Firms Quarterly Fed St Louis database 1953:I-2014:IV Effective Federal Funds Rate Monthly Fed St Louis database 1954:II-2014:IV Moody’s (30 year) BAA - AAA Corporate Bond Spread Monthly Fed St Louis database 1953:I-2014:IV Financial Business Total Liabilities: Net Worth Quarterly Fed St Louis database 1953:I-2014:IV Nonfinancial Corporate Business Net Worth Quarterly Fed St Louis database 1953:I-2014:IV GDP Deflator Quarterly Fed St Louis database 1953:I-2014:IV
Note: All variables were downloaded on March 2, 2015. The latest recorded observation for each variable was 2014:IV (except for capital stock).
1
2.1.1
Correlations
2
Let us begin with the correlations reported in Table 2. The DSGE model is capable of
3
matching a number of correlations such as the autocorrelations of output, inflation,
4
and (especially) capital. The matching of demand side correlations is also very good.
5
The correlation of investment to output, and consumption to output is almost the
6
same as the value obtained from the US data (0.98 vs 0.90 and 0.30 vs 0.32). A
7
similar accomplishment is also achieved for the stock market variables in the model.
8
The correlation of stock prices to output is 0.97 in the model and 0.83 in the data.
9
Equally, residual earnings to output has a correlation of 0.61 in the model and 0.76
10
in the data. Lastly, the book value to output has a correlation of 0.59 in the model
11
and 0.90 in the data.
12
It also manages to capture some of the supply side correlations. In particu-
13
lar, the data correlations of capital-output, marginal costs-output, capital-interest
14
rate, capital-marginal costs, and labor-output are closely matched by the model.
15
Moreover, and confirming the observations in the impulse response analysis, the
16
correlation between the front payment share for capital purchases ϑt and output is
17
strongly countercyclical (-0.97). Another interesting insight comes from the corre-
18
lation between the share, ϑt and capital (-0.31), or residual earnings (-0.67). These
19
numbers are in line with the intuition from the model. A higher (lower) pre-payment
20
share will force entrepreneurs to use a higher (lower) proportion of their liquid funds
21
to fund their capital purchases. However, since these are limited, the total amount
22
of capital they can purchase will be lower (higher) compared to the case without
20
Table 2: Model correlations - comparisons Correlations Value - behavioural model ρ(yt , yt−1 ) 0.85 ρ(yt , kt ) 0.67 ρ(yt , πt ) -0.46 ρ(yt , St ) 0.80 ρ(yt , ast ) 0.80 ρ(yt , ADt ) 0.15 ρ(yt , ASt ) -0.12 ρ(yt , ψ(ut )) -0.1 ρ(yt , dt ) 0.38 ρ(yt , lts ) 0.25 ρ(yt , rt ) 0.37 ρ(yt , it ) 0.24 ρ(yt , ct ) 0.21 ρ(yt , st ) 0.28 ρ(yt , xt ) -0.39 ρ(xt , ϑt ) 0.01 b 0.03 ρ(yt , nt ) ρ(yt , ht ) 0.84
Value - DSGE model 0.59 0.26 0.83 0.97 0.61 −1 ∗ 10−3 0.78 0.98 0.30 0.46 -0.84 0.8 0.97
Value - US data 0.85 0.15 -0.43 0.83 0.76 0.32 0.18 0.45 0.90 0.32 -0.28 -0.49 0.45 0.88
Correlations ρ(yt , ϑt ) ρ(kt , kt−1 ) ρ(kt , ast ) ρ(kt , ϑt ) ρ(kt , rtn ) ρ(lts , kt ) ρ(lts , xt )
Value - behavioural model -0.02 0.94 0.51 2 ∗ 10−3 0.15 0.27 -0.1
Value - DSGE model -0.97 0.95 0.81 -0.31 0.49 -
Value - US data 0.88 0.31 0.38 0.26
ρ(πt , πt−1 ) ρ(πt , ast ) ρ(πt , rtn ) n ) ρ(πt , rt−1 ρ(xt , xt−1 ) ρ(xt , ast ) ρ(xt , kt ) ρ(ϑt , ast ) ρ(yt , nft ) ρ(yt , qt )
0.79 -0.44 0.56 0.51 -0.01 -0.12 -0.27 0.07 0.02 -
0.41 0.13 -0.91 -0.1 -0.23 0.07 -0.67 0.97 0.59
0.93 0.34 0.34 0.68 0.09 0.22 0.90
Note: GDP deflator was used as the inflation indicator, 3-month T-bill for the risk-free interest rate, the deposit rate as the savings indicator and the Corporate lending risk spread (Moody’s 30-year BAA-AAA corporate bond rate) as the counterpart for the firm borrowing spread in the models. The variables that are left blank do not have a direct counterpart in the data sample. These are also called ’deep variables’. The only way is to estimate a structural model (using for instance Bayesian techniques) and to derive a value based on a (theoretical) structure. Alternatively, one could also approximate values using micro data. However, this is outside the scope of this paper.
such a constraint. Hence the negative correlation since less (more) capital will be
1
bought and accumulated in total. Along the same lines, a positive (negative) resid-
2
ual earnings means that the economic outlook of the future is positive (negative)
3
since the market value is above (below) the book value. Knowing this, CGP will
4
have less (more) doubt of entrepreneurs repayment status, and therefore will ask for
5
a lower (higher) share.27
6
Nevertheless, there is room for improvement in the data matching. More specif-
7
ically, the correlation of output to inflation is positive and high in the model (0.83)
8
while it is negative in the data (-0.43).28 Equally, the correlation between inflation
9
and the interest rate is highly negative in the model (-0.91) while it is positive in
10
the data (0.34). This implies that the correlation between output and the policy
11
rate is positive and large (0.78), but since inflation and output have the ’wrong’
12
sign in the model, the relation between inflation and interest rate is also incorrectly
13
captured. To conclude this section, the DSGE model does a good job in matching
14
a large portion of the US correlations. The supply-side and demand-side relation
15
are correctly matched, and the autocorrelations are much closer to the data than
16
in many other financial friction models. However, there is space for improvement,
17
in particular in capturing the ’true’ relation between output and inflation, and in
18
bringing the autocorrelations even closer to the data.
19
27
CGP know that during booms (busts), the probability of default of entrepreneurs will shrink (rise) and they will receive more (less) external funds in the next period, which assures CGP of receiving the full payment for their capital sold. 28 One reason for why it is negative in the data is because the relation is mainly driven by supply-side factors.
21
Table 3: Second moments - comparison Variable yt πt kt xt ast dt lts rtn it ADt
Behavioural model DSGE model 2.47 3.08 0.33 0.27 0.38 0.39 1.07 0.50 0.13 0.98 1.60 2.61 0.91 0.02 0.26 5.03 0.2 -
US data 0.016 0.50 1.50 0.18 5.68 1.36 3.55 0.76 3.08 -
Variable ϑt ct st nbt nft St qt ht ψ(ut ) ASt
Behavioural model 79.4 0.25 0.25 4.77 79.4 1.33 0.21 0.21
DSGE model 4.38 0.27 3.16 1.23 1.02 1.15 0.16 -
US data 0.81 8 1.32 2.21 10.33 2.01 1.18 -
Note: The moments are calculated taking output as the denominator. Following a standard approach in the DSGE literature, this is in order to examine the moments with respect to the general business cycle.
1
2.1.2
Standard deviations
2
To continue with the (relative) standard deviations reported in Table 3, the DSGE
3
model captures the qualitative nature of cycles in the data. With the only exception
4
of capital and residual earnings, the model attributes the right type of (relative)
5
standard deviation for all the other variables. In other words, when a variable is
6
more volatile than the observed GDP (business cycle) in the data, we obtain the
7
same result in the model, and vice versa. In addition, for variables such as inflation,
8
investment, labor, and net worth of firms, the (relative) standard deviations in the
9
model are very close to the numbers in the data. Taking into account that we did
10
not explicitly follow the procedure of ex ante moment matching when calibrating
11
the model, the results are very promising.
12
Where the model could do better is in replicating the second moments of capital
13
and residual earnings. Whereas capital and residual earnings are more volatile than
14
the business cycle in the data, they are less volatile in the model. This implies
15
that their responsiveness to shocks over the business cycle is higher in the data.
16
Also for some variables, such as the stock market price, consumption or the (policy)
17
interest rate, the standard deviations in the model could be increased in order to
18
approximate the empirical figures, even if their general business cycle characteristic
19
is correct. Taken altogether, however, the DSGE model is effective in replicating
20
most of the data volatilities, even better than for correlations, and without having
21
to adopt abstract ad hoc modelling tricks, such as including a long list of (large)
22
shocks, or introducing autoregressive structures in key model variables.
22
2.2
Variance decomposition
1
Next, we would like to apprehend the most important shocks for explaining the
2
variation in the model. To do so, we decompose the volatilities of all variables using
3
the five shocks we have introduced earlier. The percentages are reported in Table 4.
4
The first observation is that the financial shock, followed by the monetary policy
5
and technology shocks explain the vast majority of the volatility in the model. The
6
asset price shock and the shock to the utilization costs are, in contrast, almost
7
irrelevant.
8
Continuing with output, more than half of its volatility is explained by the fi-
9
nancial shock. Approximately a fourth is explained by the monetary policy shock,
10
and just under a fifth by the technology shock. That is not surprising since the
11
financial accelerator mechanism plays a significant role in the model construction.
12
What is more surprising and affiramative of the importance of the interaction be-
13
tween the supply-side and finance is that taken together, the TFP and financial
14
shock roughly explain three-fourths of the variation in the majority of the model
15
variables.29 Taking further into account that a monetary policy shock is transmitted
16
via demand side (via consumption Euler equation and investment demand) as much
17
as via the supply side (via cost of capital, intertemporal risk smoothing in capital
18
input market, and more broadly firm marginal production costs), it is clear that the
19
supply side in conjunction with financial frictions is the most important motor in
20
the model. Moreover, it suggests that the current model is sufficiently different from
21
the canonical BGG (1999) or Gerba (2014), in which the demand side coupled with
22
financial frictions explained the largest part of the variation.30
23
2.3
24
Impulse response analysis in the DSGE model
To maintain the focus, we will only discuss the TFP and financial shocks in this
25
section. Note that the numbers on the x-axis indicate the number of quarters.
26
We will also include a comparison of the results in the current framework with 29
The only exceptions are the real interest rate, the utilization cost function and marginal costs. For a variance decomposition of the BGG (1999) and Gerba (2014) models using the same calibrations, do not hesitate to contact the authors for details. 30
23
27
Table 4: Variance decomposition - DSGE model (in percent) Variable Shock to firm financing costs Monetary Policy shock yt 56.85 26.12 ct 10.65 38.09 it 70.08 21.72 nft 69.85 24.35 rkt 75.8 17.98 rt 3.85 95.91 rtn 73.43 21.67 kt 70.29 21.26 ψ(ut ) 0 0 qt 69.01 20.36 xt 39.76 56.73 ϑt 70.15 23.35 ht 64.46 30.17 πt 57.74 38.72 st 70.08 21.73 ret 65.65 20.82
Technology shock 16.68 51.23 0.40 5.51 5.83 0.22 4.72 8.17 0 7.81 2.33 6.23 4.86 3.25 7.80 7.83
Asset price shock Utilization cost shock 0.35 0 0.03 0 0 0 0.28 0 0.39 0 0.01 0 0.19 0 0.28 0 0 100 2.82 0 0.31 0.88 0.27 0 0.40 0.12 0.25 0.24 0.40 0 5.70 0
Note: The variance decomposition is calculated on the simulated paths of the model variables when all five shocks are simultaneously (but independently) activated.
1
model versions where the stock market and supply side financial frictions are omit-
2
ted (original BGG, 1999), or a version where only the stock market mechanism is
3
included (Gerba, 2017). For sake of comparability, we have calibrated the aforemen-
4
tioned models in the same way as for the current model, as well as applied the same
5
shock structure and parametrisation.
6
2.3.1
7
Figure II.1 depicts the responses to an expansionary TFP shock. An increase in
8
productivity of 0.5% results in an expansion in production and sales, which increases
9
the profitability of the firm. This pushes up its market value by 0.8%, its net worth
10
by 1.75%, marginal costs down by 0.2% and labour demand up by 0.8%. Moreover,
11
because the profitability outlook of the firm is positive, the CGP become less worried
12
about the repayment of their capital sold, which relaxes the down payment share
13
by 2.5% (i.e. less of the capital purchased has to be pre-paid). The book value
14
of the firm also goes up by 0.5%. However, since the profitability of the firm is
15
expected to remain high for multiple periods ahead, and its investment demand
16
is highly positive and stable (initially at 3.5%, and then 0.8% after 16 quarters),
17
market value should be higher than the book value for several periods ahead. That
18
is exactly what we observe in residual earnings, which increase by up to 0.7% in
19
quarter 4. The effects on inflation are, however, non-standard. While a positive
20
TFP shock reduces inflation, the demand effects from an expansion in investment,
Productivity shock
24
stock prices, consumption (0.5%) and external financing are so strong that they
1
offset the initial fall, which results in a final increase of inflation by 0.04%. This
2
triggers a positive (albeit marginal) increase in the policy rate of 0.005%, resulting
3
in a very short-lived inflation. Therefore, the real rate falls (-0.018%). Lastly, the
4
total effect of this supply-side expansion is that output expands by 1%, and remains
5
above its steady-state level for multiple quarters.
6
To quantify the importance and the propagating power of the mechanism devel-
7
oped in this paper, we briefly compare the responses in the current model with the
8
benchmark BGG (1999), and the Gerba (2017) extension. Judging from the dotted
9
lines in Figure II.1, one can clearly see that by omitting a (stock) market valuation
10
mechanism of firms and an explicit interaction between supply-side and financial
11
markets, the expansionary effects from a TFP shock are considerably smaller. Net
12
worth of firm increases 3 times less (or 0.55%), marginal costs drop 50 times less
13
(or -0.004%), and contrary to above, labour demand falls (since the TFP effect is
14
entirely on employing capital more effectively). As a result, the (book) value of firm
15
increases by less than a half (or 0.2%) and investment demand more than 3 times
16
less (or 1%) compared to the full model. Because of this weaker transmission in the
17
canonical BGG (1999) model, inflation falls instead (by 0.0015%), which pushes the
18
policy rate down and the real rate up, and output increases by 50% less (or just
19
0.5% above the steady state level).
20
Even when we include the (stock) market valuation mechanism, as in Gerba,
21
2017, the responses are weaker. More specifically, while the responses in Gerba
22
(2017) and this model are qualitatively the same, the magnitudes of the responses
23
in the current model are, on average, 15% higher.31
24
2.3.2
25
Financial shock
The second shock we consider is a 0.5% reduction in the cost of external financing
26
for firms. The impulse responses are depicted in Figure II.2. The immediate impact
27
is that the return on capital rises by 3% and the real rate falls by nearly 0.1%,
28
31 For the sake of space, we have not reported the impulse responses of the Gerba (2017) model. However, should you wish to see them, please do not hesitate to contact the authors.
25
1
which makes borrowing and investment much more attractive for entrepreneurs.
2
They will therefore borrow up to the new maximum, and increase their investment
3
by 10%. Moreover, firms will produce more since their marginal costs have gone
4
down by 0.8% and their demand for labor up by 2.3% as a result of an increased
5
production capacity. Therefore capital also increases by 1.1%. Higher production
6
and investment implies a higher net worth in the future, which eventually increases
7
by 6.5%. As a result of the positive outlook on firm finances and its realized cash-
8
flows, both the market and book value of firms increase. The market value rises
9
by 2.5%, meanwhile the book value by 1.8%. Since the expectations of future firm
10
profits and investment returns are high, these are additionally priced in today’s
11
market value, which at the peak (after 4 quarters) pushes the residual earnings up
12
to 1.75% above the steady-state level.
13
Note the (positive) supply side effects that a (positive) financial shock has in
14
this framework. Not only does it increase production, reduce the marginal input
15
costs, and increase the market value of firm, but it also relaxes the front payment
16
share by 9% since CGP are less worried about entrepreneurs repayment-status, and
17
therefore require smaller pre-payment. All of this results in an output increase of
18
2%. That is twice the expansion originated from a supply shock only.
19
In comparative terms, the expansion is attenuated in the canonical BGG (1999)
20
version. In it, return on capital increases by 1% and the real rate falls by 0.015%.
21
The consequence is a much smaller (positive) borrowing gap, and so the increase in
22
investment is 4 times smaller than the one observed in the full model (i.e. 2.5%).
23
In addition the rise in production is also significantly smaller. The fall in marginal
24
costs is 5 times smaller (0.15%) and the rise in the demand for labor 4 times smaller
25
(0.6%). As a consequence, net worth increases only by 2.25% and the (book) value
26
by 0.6% (a third of the values in the full model). The resulting inflation rise is
27
6 times smaller, the same as for the real interest rate (0.03% and -0.015%)32 The
28
aggregate effect on output is that it increases by 0.45%, or by less than a fourth to
29
the full model.
30
Including a (stock) market strengthens the (financial) shock transmission mech32
Logically, the fall in consumption is also smaller, 0.05%, as the real rate falls by less.
26
anism, even if less significantly than in the full model. The impulse responses are
1
on average 25% lower in the Gerba (2014) extension compared to the model here.
2
These observations imply that including the interaction between stock markets, ex-
3
ternal financing and the supply side does not only amplify the shocks, but that the
4
amplification is stronger when the economy faces a financial shock compared to a
5
real shock only.33 Moreover, under the current framework financial shocks which are
6
predominantly transmitted via the supply side have stronger macroeconomic effects
7
than those transmitted predominantly via the demand side (see online appendix).
8
That is very much in line with the empirical observations outlined earlier in the
9
introduction. Aggregate supply is a powerful propagator of financial shocks in the
10
current model.
11
2.4
12
Recap of findings: DSGE model
Qualitatively, the full model represents a significant improvement in moment match-
13
ing and impulse responses compared to other streamlined versions. The model does
14
a decent job in matching a large share of the correlations in the US data. In par-
15
ticular, the standard supply-side and demand-side relations are correctly matched
16
in this model, and the autocorrelations are much closer to the data than in many
17
other financial friction models. The performance of the model is even better for
18
volatilities. The full DSGE model if effective in replicating the qualitative aspects
19
of most of the data volatilities without having to recur to a long list of (large) shocks
20
or by introducing high autoregressive structures on key model variables. Further,
21
supply side financial frictions are indeed a powerful accelerator mechanism. Our
22
variance decomposition exercise shows that taken together, the TFP and financial
23
shock roughly explain three-fourths of the variation in the majority of the model
24
variables. This result is sufficiently different from any of the other streamlined ver-
25
sions of the model, where the demand and financial shocks explain the largest share
26
of the variation. Along the same lines, the complete feed-back mechanism intro-
27
duced in this paper is indeed a more powerful propagator of shocks than any of the
28
other versions of the model where that has only partially been captured. In general,
29
33
See online appendix for further analysis of real-financial shocks.
27
1
the impulse responses to real or financial shocks are between 15% and 5 times larger.
2
Even more interesting, there are heavy supply side effects from financial shocks in
3
this framework. To put it into perspective, the full feed-back mechanism between
4
asset prices, financial sector and production is so strong that financial shocks have
5
4 times larger impact on output and other aggregate variables compared to a stan-
6
dard financial accelerator model (BGG, 1999), or as much as 25% larger than the
7
augmented stock market version.
8
2.5
9
As for DSGE, we begin with an (ex post) moment matching exercise of the (model
10
generated) moments to the US data. A full list of variables and other details can be
11
found in Table 3.
12
2.5.1
13
The behavioural model precisely matches the correlations of many supply-side and
14
financial variables. This includes credit to firms, deposits, the (risk-free) interest
15
rate, inflation, and firm financing spread. It is also very successful in reproducing
16
the autocorrelations of output, capital, and inflation, as well as the correlations
17
between capital and credit to firms, and inflation and the (risk-free) interest rate.
18
However, there is room for improvement in matching stock variables, such as firm
19
and bank net worths, some macroeconomic aggregates (investment mainly) as well
20
as the autocorrelation of firm financing spread. While they are all acyclical and
21
not persistent in the model, they are highly procyclical and highly persistent in the
22
data.
23
2.5.2
24
Turning to (relative) second-, third-, and fourth moments in Table 5, the model
25
is highly successful in reproducing the moments of inflation, the (risk-free) interest
26
rate, credit to firms, deposits, and net worth of banks. It is also successful in making
27
net worth of firms more skewed and more leptokurtic than output. However, the
Moment matching in the behavioural model
Correlations
Second and higher moments
28
Table 5: Higher moments - Behavioural vs data Variable Skewness behavioural yt -1.66 πt -0.009 kt 0.31 xt -9.24 ast 0.02 dt -0.05 s lt 0.16 1.27 rtn ADt -0.03
Skewness data -0.42 -0.66 0.82 -5.8 1.53 1.36 -0.61 -1.27 -
Kurtosis behavioural Kurtosis data Variable 15.94 0.22 ϑt 0.25 3.54 ψ(ut ) 0.36 -1.66 ct 20.52 58.6 st 0.12 27.15 nbt 0.18 4.54 nft 0.13 3.57 St 1.1 2.38 it 0.19 ASt
Skewness behavioural 15.83 -0.001 -4.61 4.61 -15.78 -15.83 -15.84 4.63 0.01
Skewness data 0.37 0.49 -2.34 -0.34 1.57 1.18 -
Kurtosis behavioural 48.38 0.03 8.78 8.8 48.19 48.37 48.43 8.68 0.19
Kurtosis data 0.14 8.39 9.39 16.37 5.18 0.71 -
Note: The moments are calculated taking output as the denominator. In the US data, moments are calculated taking real GDP as the denominator. These are calculated using the full sample of US data stretching from 1953:I - 2014:IV. During this period, the US economy experienced 10 cycles (using NBER business cycle dates), and the average GDP increase per quarter during expansions was 1.05% while it was -0.036% during recessions. The data were de-trended using a standard two-sided HP filter before the moments were calculated in order to facilitate comparison with the model generated (cyclical) moments. The variables that are left blank do not have a direct counterpart in the data sample. These are also called ’deep variables’. The only way is to estimate a structural model (using for instance Bayesian techniques) and to derive a value based on a (theoretical) structure. Alternatively, one could also approximate values using micro data. However, this is outside the scope of this paper.
moments of the latter are higher in the model compared to US data. On the other
1
hand, capital and investment are smoother in the model.
2
Another strength of the model lies in reproducing irregular business cycles. In
3
contrast to standard first-, second-, or even third order approximated DSGE models.
4
the behavioural model generates substantial asymmetries between expansions and
5
recessions as well as produces non-Gaussian probability distribution functions for
6
most variables. That is much more in line with the observed pattern in the US
7
cyclical data. These observations are not only in line with the patterns found in the
8
US (Fagiolo et al, 2009), but also for other OECD countries (Fagiolo et al, 2008).
9
Nonetheless, for some variables (net worth, consumption, savings, (risk free) interest
10
rate, and credit to firms) the model generates excessive skewness and/or kurtosis.
11
Note that standard log-linearized DSGE models are not capable of mimicking non-
12
normality in the distribution of the output gap.
13
To sum up, the model matches a number of crucial US data variables. This
14
includes supply-side and financial variables such as the (risk-free) interest rate, in-
15
flation, credit to firms, deposits, firm financing spread and net worth of banks. It
16
is also successful in matching several supply relations (capital-firm credit, inflation-
17
interest rate) as well as their autocorrelations (output, capital and inflation) There
18
is, however, some scope for improvement in matching demand-side variables (such
19
as consumption, savings, investment) as well as stocks (net worth of firms).
20
2.6
21
Frequency domain
A valid criticism of the above discussion is that the probability distribution functions
22
(pdf:s) and moments generated for those variables is not unique and multiple time
23
29
1
series can generate similar (or same) statistics. Expressed differently, multiple data
2
generating processes can result in the same pdf:s and moments.
3
To overcome this problem, we examine the data generated in the behavioural
4
model in the frequency domain, instead of the usual time domain.34 In particular,
5
we simulate Welch’s overlapped segment averaging estimator for all the variables
6
in the model and examine the estimators of the second order spectrum. Briefly,
7
Welch averaged periodogram divides the (frequency) spectrum into longest possible
8
sections, each windowed with a Hamming window. The Welch estimator tells you
9
what frequency/frequencies are dominant in your signal. This method is used for
10
data with noise. Most time-series fall within this category.
11
The power spectral densities of the standard economic variables are reported in
12
Figure II.7, while the densities for the novel variables in this model are reported
13
in Figure II.8. The number of simulations for each variable is set to 1000 and the
14
domain window is set to [500,300,500]. The length of the signal is set to 500 discrete
15
Fourier Transform points to fit the bins. 300 of those 500 are set to be overlapping.
16
Lastly we report the 95% confidence interval for each variable in order to get a
17
clearer picture of the full spectrum.35
18
The majority of the standard macroeconomic variables in Figure II.7 have a
19
AR(1) process with the autoregressive factor (ρ) between 0.5 and 0.8. So for in-
20
stance, output is an AR(1) with a high ρ of 0.9. The same process can be deduced
21
for interest rate, inflation, capital and deposits, but with a slightly lower ρ (0.8 for
22
interest rate, 0.7 for inflation, 0.6 for capital and 0.5 for deposits). Deposits also have
23
a similar autoregressive process to capital. Consumption, on the other hand, seems
24
to be a white noise process with a long-memory.36 Since this is a closed-economy
25
model, investment is equal to savings and therefore the two power spectral densities
26
look identical. Both seem to be a white noise with 1 seasonal pattern. However, the
27
white noise is a ’nicely behaving’ one as the spectrum almost looks like an inverse 34
We do not perform the same analysis for the DSGE model since the distributions of the variables are Gaussian due to the log-linearization of the model. Hence the frequency domain analysis becomes trivial in that case. 35 Note that for a time series, the frequency reported in the graphs can be viewed as the number of periods, where frequency=1/periods. 36 Remember that consumption is not explicitly modelled, but is a residual of agents’ savings decision and therefore is not expected to have a specific process in the current model set-up.
30
AR(1) process.37
1
Turning to the supply side and financial variables, we see a much more varied
2
picture. The supply side variables such as utilization costs and the share of pre-
3
payment in the input market are white noise processes with 2 seasonal peaks for
4
utilization costs. Note also that stock market prices also resemble a white noise,
5
which is in line with several studies that show that asset prices can best be modelled
6
as random walk (Fama and French (1988) or Poterba and Summers (1998)). On the
7
other hand, animal spirits, loan supply and loan demand all exhibit characteristics
8
of an AR(1) process with a ρ close or equal to 0.5. As a result, we are able to
9
capture the persistence in animal spirits in the time as well as frequency domains.
10
To conclude, the deep variables of the behavioural model show persistence over time,
11
with either some seasonality patterns or long memory. On the other hand, variables
12
which other empirical studies have found to follow a (near) random walk, such as
13
stock prices, utilization costs or the pre-payment shares are also white noises in our
14
model.
15
2.7
Impulse response analysis in the behavioural model
16
Figure II.3 depicts the (median) impulse responses to a TFP shock amongst a dis-
17
tribution of impulse responses generated with different intialisations (or realization)
18
of shocks. Figure II.4 does the same for a (negative) shock to firm external financing
19
conditions.38 This shock is well representative of the pre-2008 period, where the firm
20
financing conditions were very lax and they were able to borrow at a previously
21
unseen low cost. Symmetrically, following the crash on financial markets in 2007-08,
22
the external financing costs spiked, and the transmission to the real economy, via
23
the supply side can be inversely interpreted from the current impulse responses.39 In
24
37
For the unaccustomed eye to the analysis in frequency domain, we can largely summarize the various time processes into the following frequency categories. A flat spectrum means that the underlying time series is a white noise. A spectrum that slowly decays from zero frequency onwards is an AR(p) process. A periodic spectrum with peaks or troughs is one that characterizes seasonality in the data. Lastly, a spectrum peaking at zero, but flat otherwise is that of a long memory process. 38 As for the DSGE model we will only discuss the impulse responses to the TFP and financial shocks. For a longer discussion of other shocks, please refer to the online technical appendix. 39 Since the transmission is symmetric for a positive or negative shock.
31
1
particular, we would like to test whether the behavioural model is capable of cap-
2
turing the financial-supply side interactions that were noted by several empirical
3
studies mentioned in the introduction.
4
Note that the numbers on the x-axis indicate number of quarters. All shocks
5
are introduced in t=100 and we observe the responses over a long period of 60
6
quarters. For the sake of clarity in the exposition, we will only concentrate on the
7
median impulse response however, which is a good representation of the overall
8
(non-Gaussian) distribution.40
9
2.7.1
Technology shock
10
An improvement in TFP of 0.5% results in an inflation reduction (1%) and a more
11
than proportional output expansion (1.15%). This is a result of both the increased
12
capacity in the final goods market, but also from an increase in investment (0.3%)
13
following the heavy fall in interest rate (1.3%) as a response to the falling inflation.
14
Following this general supply-side expansion, deposits and loans to firms also in-
15
crease (1 and 1.3% respectively) since the value of firm net worth (i.e. collateral)
16
has increased. As a consequence of the lower marginal cost to investment and higher
17
marginal return on capital, capital accumulation increases significantly in the next
18
period (0.5%). This results in an overall optimism on the market (animal spirits
19
rise by 0.1%).
20
However, as soon as the inflation starts recovering, interest rate react very rapidly
21
to their increase and start rising (0.35%). Because of this rise in cost of capital,
22
coupled with the fall in external financing for firms, investment and output expansion
23
reverts. However, unlike in the DSGE models, the model has eventually reached a
24
new steady state, where bank loans, deposits and equity are permanently 1.1%, 0.7%
25
and 0.1% above the previous pre-shock level.
26
in the behavioural model will have long-lasting positive effects on the banking sector 40
41
Hence a temporary technology shock
Keep in mind, when interpreting the results, that the impulse responses in the DSGE model are not conditional on the realization of shocks, since no learning occurs, and thus only a ’representative’ unconditional IRF is depicted. 41 In DSGE models, this is only possible to achieve with permanent or continuously inserted shocks.
32
and financial efficiency.42
1
2.7.2
2
Financial shock
A relaxation in the external financing costs for firms means that they will be able to
3
increase their borrowing by 0.2%, and thus their leverage. The same is true for banks,
4
since they reduce their equity by 0.1% in order to increase their lending to firms.
5
Firms will use this new credit to increase their investments by 0.2%. Production will
6
also increase, which will push firm net worth up in the future. This positive outlook
7
produces optimism in the market, generating an increase in animal spirits of 0.6%.
8
This acceleration in activity pushes output and inflation up by 0.4% and 0.035%.
9
Monetary authority is rapid in responding to the rise in inflation and raises the policy
10
rate by 0.33%, with the desired consequence of attenuating the initial expansion
11
to bring output and inflation back to their pre-shock level after approximately 3
12
years (or 12 quarters). Note that, in contrast to the case with supply shocks, the
13
financial market variables (loans, deposits and bank equity) return to their pre-
14
shock level relatively swiftly.43 We believe the reason lies in the model construction.
15
Since alterations in the cost of corporate financing are transmitted via demand-
16
side channel in this model, the macroeconomic effects are short-term and there is
17
therefore no fundamental reason for why credit should be supplied at a new level. On
18
the other hand, when the economy is faced with supply shocks, the macroeconomic
19
impact is more long-lasting, and the bank can therefore provide more (less) credit
20
at the higher (lower) productivity level. This endogenous mechanism is very much
21
in line with what has been argued in the empirical macroeconomic literature that
22
fundamental changes in the real economy will be reflected in permanent changes
23
in the financial sector activity.44 Furthermore, notice that this mechanism is very
24
difficult (if not impossible) to capture in the current generation of DSGE models
25
unless permanent shocks are introduced.
26
42
Arising from the additional dynamics generated by learning. For a monetary policy shock, the financial market variables reach a new level following the shock, but at a much lower magnitude than any of the supply side shocks. 44 For instance, think about the effects from oil shocks on the subsequent deregulation in, and expansion of the financial sector, or the IT-revolution on the long-term quantity of credit supplied and the banks’ balance sheet expansion. 43
33
1
2.8
Recap of findings: Behavioural model
2
The behavioural model has a good statistical or empirical performance. It pre-
3
cisely matches the correlations of many supply-side and financial variables such as
4
capital-firm credit, inflation-interest rate, or autocorrelations of output, capital and
5
inflation. Moreover, the model replicates the second-and higher moments of many
6
variables found in the US data such as firm financing spread, credit to firms, interest
7
rate, inflation, or net worth of banks. Even when examined under frequency domain,
8
the pattern of the (model-generated) cycles is very close to those found in the data.
9
The deep variables of the behavioural model show high persistence over time, with
10
either some seasonality patterns or long memory. On the other hand, variables that
11
other empirical studies have found to follow a (near) random walk, such as stock
12
prices, utilization costs, or the pay-in-advance share are equally white noises in our
13
model. Another strength of the model lies in reproducing irregular business cycles
14
and non-Gaussian probability distribution functions for most variables. This is in
15
line with the empirical patterns found for the US (Fagiolo et al, 2009)and other
16
OECD countries (Fagiolo et al, 2008).
17
On the qualitative end, we find that structural shocks, such as productivity have
18
a strong impact on the economy in this model. In particular, a temporary (positive)
19
technology shock in the behavioural model will have long-lasting positive effects on
20
the banking sector and financial efficiency. This is because ceteris paribus banks can
21
provide more credit to all sectors at the higher productivity level. Shocks that are
22
transmitted via demand-side (such as the shock to the cost of corporate financing)
23
have, on the other hand, only short-term effects and thus do not result in a new
24
level in credit supply and financial efficiency.
25
3
26
DSGE versus behavioural: a careful comparison
27
Now that we have completed the individual model analyses and examined the role of
28
the full-fledged feed-back mechanism in both frameworks, we are in position to focus
34
our attention on the function played by beliefs and heterogeneous expectations for
1
the efficacy and performance of the mechanism introduced in this paper. Simultane-
2
ously, the same exercise will serve the purpose of an empirical/statistical validation
3
of both models, and a methodological evaluation regarding which framework comes
4
closest to the patterns (or stylized facts) found for the US. Once again remember
5
that the endogenous mechanisms and shocks are equivalent (to the extent possible)
6
in both models. As a rough verifier of this synchronicity between the models, notice
7
in Table 3 that the persistence as well as the amplitude of the business cycles are
8
very similar (0.85 vs 0.59 and 2.47 vs 3.08). The only model characteristic that sep-
9
arates the two lies in agents’ information set and expectations formation. We will
10
proceed in three steps. First, we will contrast the second moments between the mod-
11
els and the data and determine which one approximates it the best. Second, we will
12
compare the impulse responses and determine in which model and for which shocks
13
the transmission power of the fully-fledged supply side financial friction mechanism
14
is the highest. To finish off, we will discuss the relevance of irregular business cycles
15
and (distributional) asymmetries for the US economy. Our main (methodological)
16
concern here is whether one needs to turn to large and computationally burdensome
17
non-linear belief type of model in order to replicate many of the stylized facts we
18
have discussed, or if a linear rational expectations version is sufficient or adequate.
19
3.1
20
Second moment matching in the two models
Starting with —textitex post matching of correlations in Table 2, both models do
21
a good job in capturing many (if not most) of the correlations. However, the be-
22
havioural model does even better and matches 13 correlations better than the DSGE,
23
while the opposite number is 4. In addition the behavioural model manages to
24
exactly match 5 of the correlations and the DSGE model 4. Roughly speaking,
25
the correlations that the behavioural model is better in capturing are the auto-
26
correlations (output, capital, inflation), the stock market cycle (stock price-output,
27
animal spirits/residual earnings-output), prices (inflation-output, real rate-output,
28
inflation-interest rate), and many of the supply-side relations (loan supply-output,
29
marginal costs-output, capital-interest rate, capital-loan supply, and labor demand-
30
35
1
output). On the other hand, the DSGE model is better in matching the capital series
2
(capital-output, capital-marginal costs and the autocorrelation of capital) and some
3
demand-side variables (investment-output, consumption-output).45
4
We find a similar pattern for the second moments in Table 3. Since in both
5
models the amplitude of the business cycles is very similar, we can be safe in di-
6
rectly comparing the relative standard deviations. In 8 cases, the behavioural model
7
matches more precisely the second moments, while the number of cases is 5 for the
8
DSGE. So the behavioural model has a comparative advantage in prices (inflation,
9
interest rate, stock market price), and some of the financial accelerator variables
10
(loan supply, net worth of banks, net worth of firms). The DSGE model, on the
11
other hand, matches more of the supply side variables (marginal costs, labor de-
12
mand) as well as some demand-side ones (investment, consumption, book value).
13
46
To sum up the empirical fit, both models do a good job in capturing the standard
14
statistical (second) moments in the US data. While the strength in the behavioural
15
framework lies in replicating the autocorrelations, the statistical moments of prices
16
(including stock prices), some stock variables, and some supply-side relations, the
17
DSGE has a comparative advantage with respect to capital and the demand-side
18
relations. Nonetheless, the behavioural framework outperforms the DSGE in the
19
total number of replicated moments.
20
3.2
21
Continuing with the impulse responses, we generally observe a stronger transmis-
22
sion and higher responses to supply side shocks in the behavioural model but to
23
financial/monetary shocks in the DSGE model. Let us begin with the TFP shock.
24
Comparing the two Figures II.3 and II.1, it seems that the TFP shock is, in rela-
25
tive terms, transmitted more heavily via the demand side onto output in the DSGE
26
model. This inference is based on the fact that while investment responds by sig-
Impulse responses in the two models
45
The only 5 correlations that neither of the models manage to replicate are savings-output, loan supply-marginal costs, autocorrelation of marginal costs, net worth of banks-output, and net worth of firms-output. 46 The only 3 standard deviations that neither of the models were capable of matching is capital, animal spirits/residual earnings, and savings.
36
nificantly more in the DSGE model (3.5% vs 0.3%), inflation rises (while it falls
1
in the behavioural), the interest rate marginally rises (while it falls heavily in the
2
behavioural) and output rises by less (1% vs 1.15%). Furthermore, the financial
3
market variables in the behavioural model converge towards a significantly higher
4
level compared to the pre-shock state. Remembering moreover that the autore-
5
gressive parameter in the DSGE model is set to 0.99 while none is included in the
6
behavioural, it implies that the supply-side transmission is much more powerful
7
in the behavioural model compared to the DSGE. For the financial shock, on the
8
other hand, the impulse responses in the DSGE model are between 5 to 10 times
9
higher. In the DSGE (behavioural) model, output rises by 2% (0.4%), inflation by
10
0.2% (0.035%), investment by 10% (0.2%), residual earnings by 1.8% (0.6%) and
11
capital by 1% (0.1%). For this shock, no autoregressive parameter has been included
12
in either of the models. Therefore in this case, the cognitive limitation of agents
13
plays a smaller role in the propagation of financial shocks, while the supply channel
14
as a financial shock propagator plays a more important role.
15
3.3
16
Business cycle asymmetries: How important?
The last point of comparison is the relative importance of including model asym-
17
metries over the business cycle. We have already seen that many of the variables
18
in the US data do not have the same amplitude during expansions and recessions.
19
However, the fundamental question is how important these are for the general busi-
20
ness cycle modelling and for understanding the core propagation mechanisms in an
21
economy? Is a symmetric approach a good approximation? Since we make use of
22
a method (linear approximation) that produces symmetric distributions (DSGE),
23
and at the same time a highly non-linear that produces asymmetric distributions
24
(behavioural), we are capable of evaluating the relative fit of linear approximations
25
to data, as well as try to provide an answer to the question of whether highly non-
26
linear (and complex) modelling tools are necessary in order to understand the latent
27
underlying structure of an economy.
28
If we look at the statistical values of the US business cycle in Table 5, the
29
series seems to be weakly skewed (skewness factor=-.042), but highly platykurtic
30
37
1
(kurtosis=0.22). Thus the business cycle is roughly symmetric, but has fat tails.
2
The kurtosis of a normal distribution is close to 3, while in the US data it is 0.22.
3
Hence the standard deviation is higher than that of a normal distribution. This is
4
in line with observations made by Fagiolo et al (2008, 2009) who find that not only
5
the US business cycle has fat tails, but also the business cycle of other advanced
6
economies. However, this is in contrast to earlier studies in the empirical business
7
cycle analysis literature, such as De Long and Summers (1984) or Blanchard and
8
Watson (1986) who find that there is not enough (statistical) evidence that US
9
business cycle is asymmetric. In addition, Stock and Watson (1998, 2003) and
10
Camacho and Perez-Quiros (2007) find that the long-run US GDP is both symmetric
11
and close to Gaussian. While the Gaussian distribution of US output is generally
12
true for the entire post-war period, it particularly characterizes the output cycle
13
since the beginning of the Great Moderation in 1984. Said that, Stock and Watson
14
(1998, 2003) note that US output may appear to be non-normally distributed if you
15
focus on any particular short-run period of the US business cycle (i.e. only Great
16
Inflation, only Great Recession, etc.). This is also in line with Fagiolo et al (2009)
17
who find that filtered series display fatter tails than Gaussian until the shifting band
18
takes on periodicities higher than 35 quarters, or higher than 13 quarters for one-
19
year bandwidth filters. Therefore, the conclusion one reaches regarding the nature
20
of US output cycle is contingent on the length of the sample or frequency spectrum
21
considered.
22
Symmetry of US GDP is easy to replicate in a linearly approximated model by
23
including autoregressive components to shocks. However, in order to replicate the fat
24
tails, the only way is to let autoregressive shocks have an unrealistically high volatil-
25
ity. In the behavioural model, on the other hand, the fat tails are easy to replicate,
26
but instead the distribution of output is leptokurtic and highly skewed.47 Capital,
27
consumption and investment are also roughly symmetric and platykurtic. In the
28
linearly approximated DSGE model the symmetry is correctly captured, but those
29
three variables are not platykurtic. In the behavioural model, on the other hand, all 47
Remember that the third and fourth moments are all reported in relative terms with respect to the general business cycle. Hence, to get the original moments, you have to add the relative one to the values reported for output.
38
three are both highly asymmetric and leptokurtic.48 Hence for these three variables,
1
a standard DSGE linear approximation is slightly preferred.
2
For the other variables, the conclusion is very different. Most of them are skewed
3
and leptokurtic. This is much easier to capture in the behavioural model. So,
4
for instance, the distribution of marginal costs is almost perfectly replicated in
5
the behavioural model. Also the distributions of financial variables (such as loan
6
supply, deposits, interest rate, net worth of firms, net worth of banks) and of prices
7
(such as inflation, stock market prices and animal spirits/residual earnings) are
8
closely characterized in the model. If anything, the asymmetry or kurtosis of these
9
variables is, in general, (much) higher in the model than in the data, even if the
10
general pattern is well captured. In these instances, therefore, it is not appropriate
11
to apply a linear approximation method since these statistical anomalies would not
12
at all be captured. Even non-linear perturbation methods recently applied in the
13
DSGE literature would struggle to accomplish these distributions without including
14
many frictions and shocks. Therefore which model to use depends very much on
15
what you are interested in examining. If the focus is on the general business cycle
16
and/or the aggregate demand, then the linearly approximated DSGE model is a
17
good option since it is tractable and easy to solve without compromising on the
18
complexity. If, on the other hand, the focus is to understand financial frictions,
19
the financial cycle, or the impact of supply-financial interactions on business cycle
20
anomalies, then the behavioural model is the obvious option.
21
As a final remark on the methodological differentiation, bare in mind that the
22
asymmetries in the behavioural model are endogenously generated from the learning
23
mechanism of the agents. The interaction between market frictions and the learning
24
set-up in the model leads to powerful propagation of shocks. In the DSGE model,
25
on the other hand, this propagation is achieved via the interaction between market
26
frictions and highly persistent shocks.49
27
48
Capital is the exception since it is almost symmetric and platykurtic. Except for the shock to financial costs and utilization costs, where no AR parameter is included, so the forceful propagation is purely generated from the endogenous model dynamics. 49
39
1
3.4
DSGE versus behavioural: A synthesis
2
To sum up the comparative section, both models perform well in matching the
3
standard (second) moments in the data, as well as generating powerful propagation
4
of shocks. Including the full amplification mechanism between asset prices, the
5
financial sector and the (aggregate) supply side improves significantly the empirical
6
fit compared to any other streamlined model versions. It also manages to explain the
7
mechanism behind the post-2008 business cycle experience described by Broadbent,
8
Massani and others. Although the same supply-side financial friction mechanism
9
has been introduced in both versions, the power of transmission of shocks differs
10
between the two models. While in the behavioural model the strongest transmission
11
occurs following a productivity shock, in the DSGE model it is with financial and
12
monetary shocks.
13
To round off, the benefit of further relaxing the rational expectations hypoth-
14
esis in this type of models is a better empirical fit of many macroeconomic and
15
financial variables. Moreover, the modelling of information processing and agents’
16
expectations formation is intuitive and requires less restrictive assumptions com-
17
pared to rational expectations. Said that, the trade-off is that the transmission of
18
financial (and monetary) shocks becomes weaker, and the model produces excessive
19
irregularity in cycles (and probability distribution functions) of the most common
20
macroeconomic indicators, such as output, capital or inflation. Lastly, the level of
21
complexity of the non-linear behavioural model means that it is computationally
22
more burdensome at the same time as the numerical solution method (or algorithm)
23
is less transparent and trackable compared to that of the DSGE model.
24
4
25
Linking aggregate production to asset prices and financial frictions in a fully-fledged
26
feed-back loop is a novel way of thinking about structural macroeconomic changes
27
and financial boom-bust cycles in the macroeconomic literature. On one hand,
28
temporary changes in productivity and other supply-side factors can have longer-
29
lasting effects in credit levels and bank-sector efficiency. On the other, the supply
Concluding remarks
40
side can become a powerful propagator of financial shocks, resulting in profound
1
changes in the macroeconomic structure. This mechanism can easily reproduce and
2
explain phenomena such as secular stagnation, anemic production levels, or price-
3
to-book ratio of asset prices below 1 for a sustained period of time, as portrayed by
4
Bank of England’s Financial Stability Review.
5
In the current paper, we reconstruct such phenomena by introducing a powerful
6
feed-back mechanism between asset prices, production and the financial sector in
7
an augmented financial accelerator model with an endogenous stock market mech-
8
anism. We measure the power of this mechanism to propagate and amplify shocks,
9
and quantify the empirical relevance of this mechanism using US data. In addi-
10
tion, we study the potential role that agents’ (imperfect) beliefs and heterogeneous
11
expectations play for accelerating this mechanism. We do this by comparing the dy-
12
namics and results from two extreme versions of a DSGE model: one where rational
13
expectations hold throughout, and the other where agents hold imperfect informa-
14
tion about the aggregate states they do not directly control, but learn about their
15
realizations using heterogeneous behavioural rules.
16
We find that the full model represents a significant improvement in moment
17
matching and impulse responses compared to any other streamlined version of the
18
financial accelerator model. It does a very decent job in matching the statistical
19
properties of the US data, including many of the supply-side and macro-financial
20
relations found in the data. Further analysis in the frequency domain of the sim-
21
ulated data confirms this good performance. Moreover, we demonstrate how the
22
fully-fledged device in the complete model works as a powerful accelerator mecha-
23
nism. In general, the impulse responses to real or financial shocks are between 15%
24
and 5 times larger. Even more interesting, there are heavy supply side effects from
25
financial shocks in this framework. To put it into perspective, the full feed-back
26
mechanism between asset prices, financial sector and production is so strong that
27
financial shocks have 4 times larger impact on output and other aggregate variables
28
compared to a standard financial accelerator model (BGG, 1999), or as much as
29
25% larger than the augmented stock market version. Moreover, structural shocks,
30
such as productivity have a strong impact on the economy in this model. In par-
31
41
1
ticular, a temporary (positive) technology shock in the behavioural model will have
2
long-lasting positive effects on the banking sector and financial efficiency.
3
To conclude, the benefit of further relaxing the rational expectations hypoth-
4
esis in this type of models is a better empirical fit of many macroeconomic and
5
financial variables. Moreover, the modelling of information processing and agents’
6
expectations formation is intuitive and requires less restrictive assumptions com-
7
pared to rational expectations. Said that, the trade-off is that the transmission of
8
financial (and monetary) shocks becomes weaker, and the model produces excessive
9
asymmetry in cycles (and probability distribution functions) of the most common
10
macroeconomic indicators, such as output, capital or inflation. Lastly, the level of
11
complexity of the non-linear behavioural model means that it is computationally
12
more burdensome at the same time as the numerical solution method (or algorithm)
13
is less transparent and trackable compared to that of the DSGE model.
14
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Appendices I I.1 I.1.1
10
11
12
Quantitative details
13
DSGE model
14
Model Derivation and Solution
15
The full model derivations can be found in the online technical appendix of this pa-
16
per. Once aggregate conditions have been derived, these are log-linearized around
17
a (non-stochastic) steady state. This means that we cannot capture nonlineari-
18
ties, such as precautionary savings, buffer-stock behaviours, or state-dependent out-
19
comes. We apply a linear approximation method to our solution which means that
20
our perturbation only works around the steady state. The system of log-linearized
21
equations is provided here:
22
47
1
I.1.2
Log-linearized model
2
Aggregate Demand:
3
Resource constraintsource constraint yt =
4
5
6
7
8
9
10
11
C I Ce e ct + it + c Y Y Y t
(I.1)
Consumption Euler equation ct = −rt + Et (ct+1 )
(I.2)
cet = nt
(I.3)
k rt+1 − rt = −ν(nt − (qt + kt ))
(I.4)
Entrepreneurial consumption
Financial accelerator
External Finance Premium ef pt = rtk − rt
(I.5)
rtk = (1 − )(yt − kt − xt ) + st − st−1
(I.6)
st = ψ(it − kt )
(I.7)
st = qt + ret
(I.8)
Return on capital
Investment accelerator
(Stock) Market value of capital
Residual earnings and formation of stock market expectations ret = ρre ret−1 + (χ)(Et [yt+1 ] + nt − Et [rt+1 ]) + ei
48
(I.9)
Aggregate Supply:
1
Cobb-Douglas production function
2
yt = a + αkt + (1 − α)ωht
(I.10)
Marginal cost function
3
1 yt − ht − xt − ct = ht η
(I.11) 4
Cash-in-advance constraint
5
Et [St+1 ]Kt+1 ≤ ϑt [Et [St+1 ]Kt+1 − Nt ]
(I.12)
6
Utilization cost function
7
ψ(ut ) = ξ0 + ξ1 (ut − 1) +
ξ2 (ut − 1)2 2
(I.13)
Approximated Philips curve
8
πt = κ(−xt ) + βπt+1
(I.14)
Evolution of State Variables:
9
Capital accumulation
10
kt = δit + (1 − δ)kt−1
(I.15)
Net worth accumulation
11
nt = γR
K k (r − rt ) + rt−1 + nt−1 N t
(I.16)
Monetary Policy Rule and Shock Processes
12
Monetary policy
13
n rtn = ρrt−1 + ζEt [πt+1 ]
49
(I.17)
1
Technology shock at = ρa at−1 + ea
2
(I.18)
Real interest rate (Fisher relation) rtn = rt − Et (πt+1 ) − ern
(I.19)
Table I.1: Parameters and descriptions Parameter
Description
Value
Calibrated C/Y I/Y C e /Y X α Ω η κ θ β δ γ R K/N ν φ ρre χ Ψ κ z ξ ξ1 ξ2 εa εrn εrk εuc ρ ζf ρa ρrk ρuc
Calibration Share of consumption in resource constraint 0.806 Share of investment in resource constraint 0.184 Share of entrepreneurial consumption in resource constraint 0.01 Marginal product in investment demand 0.99 Gross markup over wholesale goods 1.10 Share of capital in production 0.20 Share of household labour in production 0.99 Labour supply elasticity 5.00 Share of marginal cost in Phillips Curve 0.086 Calvo pricing 0.75 Quarterly discount factor 0.99 Depreciation rate 0.025 Survival rate of entrepreneurs 0.973 Steady state quarterly riskless rate 1.010 Steady state leverage 2.082 Elast. of EFP to leverage 0.092 Elast of inv. demand to asset prices 0.25 AR parameter on residual earnings 0.67 Parameter on the expected state of the economy in the residual earnings equation 0.18 Adjustment cost function in investment 0.5 Adjustment cost in investment parameter 11 Technological development parameter 0.5 Parameter 1 in the utilization cost function 0.8 Parameter 2 in the utilization cost function 0.3 Parameter 3 in the utilization cost function 0.25 Std. deviation of technology shock 0.5 Std. deviation of nom. interest rate shock 0.5 Std. deviation of financial shock 0.5 Std. deviation of shock in the utilization cost function 0.5 AR parameter in monetary policy rule 0.95 MP response to expected inflation 0.20 AR parameter of productivity shock 0.99 AR parameter in financial shock 0 AR parameter in shock to utilization costs 0
3
I.1.3
Calibrations and simulations
4
Table I.1 reports the full list of calibrated parameters. Most of these are calibrated
5
following the values given in BGG (1999), and are standard to the literature. There
6
are only a few minor differences. Our consumption-output ratio in the steady state
7
includes both the private and public consumption, hence why the value is slightly
50
larger in our calibration.50 We calibrate the share of capital in production, α to 0.20.
1
For robustness purposes, we also tried with α = 0.30, α = 0.35, the other common
2
values in the literature, but no noticeable differences were observed. Finally, in order
3
to replicate the stylized facts of the asset price wedge (including the market and
4
book values) of Gerba (2014), we parameterize ν, the elasticity of EFP to leverage
5
to 0.13. It is slightly higher than the 0.05 in the original BGG model, but follows the
6
estimation results for the US of Caglar (2012), and it represents well the post-2000
7
period, when the leverage of firms increased drastically, and so the sensitivity of
8
financial lending rates to leverage was high.51 In the same manner, we consider an
9
accommodative monetary policy, thus replicating the Fed’s stance during most of
10
the past decade, and use the Taylor rule parameters of 0.2 for the feedback coefficient
11
on expected inflation, ζ along with a value of 0.95 for the smoothing parameter.
12
Borrowing from the insights in the corporate finance literature, and the US
13
estimation results for the residual earnings process of Caglar (2012), we set the
14
value of the autoregressive process of residual earnings equal to 0.67. Lastly, the
15
weight on expected evolution of the economy is 0.18.
16
Following Christiano et al (2005), Smets and Wouters (2003, 2007) and Gerali
17
et al (2010), we set the capital depreciation rate δ to 0.025. The elasticity of the
18
capital utilization adjustment cost function ψ(it ) is parametrized to 0.5 as in Smets
19
and Wouters (2007).52
20
To conclude, the parameters of the function determining adjustment costs for
21
capacity utilization (ξ0 , ξ1 , ξ2 ) are set to (0.8, 0.3, 0.25) in order to capture the es-
22
timation results of Smets and Wouters (2005) who find that the capital utilization
23
costs are between 0.14 and 0.38 (Euro Area 1983-2002) and 0.21 and 0.42 (US 1983-
24
2002), with a mean of 0.25 (Euro Area) and 0.31 (US). If we normalize ut to 1 (as in
25
Christiano et al (2005), Miao et al (2013) or Auernheimer and Trupkin (2014)), then
26
the cost for utilizing capital will be 0.20 (1 − ξ0 ), which is well within the estimated
27
50
In the canonical BGG (1999) model, the C/Y ratio is calibrated to 0.568. However, if we also include the public consumption in that ratio, which they calibrate to 0.2, the value is almost the same to our, which we calibrate to 0.806. 51 See Gerba (2015) on the balance sheet changes and the financial exposure that firms underwent during the past decade. 52 This is equivalent to setting a κi equal to the estimated range (10.18 − 12.81) as in Gerali et al (2010).
51
1
intervals of Smets and Wouters (2005).
2
The standard error of all shocks is, for reasons of comparability with the be-
3
havioural model, set to 0.5. The autoregressive components of the various shocks
4
are set to standard values in the literature For the monetary policy shock, it is set
5
to 0.90 and for technology shock to 0.99. For the financial shock, and the shock to
6
utilization costs, we set the AR-component to 0 and only consider a 1-period white
7
noise shock. This is because we do not find convincing evidence in the literature for
8
incorporating a persistence parameter in these shocks.
9
I.2
Behavioural model
10
I.2.1
Expectations formation and learning
11
Next, we wish to characterize the information friction and heterogeneous expecta-
12
tions used in this model. We will focus on two things. First, we will model the
13
cognitive limitation that agents have regarding the aggregate states, which is out of
14
direct control for them. Second, we will outline the learning framework that they
15
use in order to gain a better understanding of those aggregate states. The learning
16
framework incorporates heterogeneous expectations in its intrinsic dynamics.
17
Under rational expectations, the expectations term will equal its realized value
18
in the next period, i.e. Et Φt+1 = Φt+1 , denoting generically by Φt any variable
19
in the model. However, as anticipated above, we depart from this assumption in
20
this framework by considering bounded rationality as in DeGrauwe (2011, 2012).
21
Expectations are replaced by a convex combination of heterogeneous expectation
22
operators Et yt+1 = E˜t yt+1 and Et πt+1 = E˜t πt+1 . In particular, agents forecast
23
output and inflation using two alternative forecasting rules: fundamentalist rule vs.
24
extrapolative rule. Under the fundamentalist rule, agents are assumed to use the
25
steady-state value of the output gap - y ∗ , here normalized to zero against a naive
26
forecast based on the gap’s latest available observation (extrapolative rule). Equally
27
for inflation, fundamentalist agents are assumed to base their expectations on the
28
central bank’s target - π ∗ against the extrapolatists who naively base their forecast on
52
a random walk approach.53 We can formally express the fundamentalists in inflation
1
and output forecasting as:
2
E˜tf πt+1 = π ∗
(I.20)
E˜tf yt+1 = y∗
(I.21)
and the extrapolists in both cases as:
3
E˜te πt+1 = θπt−1
(I.22)
E˜te yt+1 = θyt−1
(I.23)
This particular form of adaptive expectations has previously been modelled by
4
Pesaran (1987), Brock and Hommes (1997, 1998), and Branch and McGough (2009),
5
amongst others, in the literature. Setting θ = 1 captures the ”naive” agents (as they
6
have a strong belief in history dependence), while a θ < 1 or θ > 1 represents an
7
”adaptive” or an ”extrapolative” agent (Brock and Hommes, 1998). For reasons of
8
tractability, we set θ = 1 in this model.
9
All variables here are expressed in gaps. Focusing on their cyclical component
10
makes the model symmetric with respect to the steady state (see Harvey and Jaeger,
11
1993). Therefore, as DeGrauwe and Macchiarelli (2015) show, it is not necessary
12
to include a zero lower bound constraint in the model since a negative interest
13
rate should be understood as a negative interest rate gap. In general terms, the
14
equilibrium forecast/target for each variable will be equal to its’ steady state value.
15
Next, selection of the forecasting rule depends on the (historical) performance of
16
the various rules given by a publically available goodness-of-fit measure, the mean
17
square forecasting error (MSFE). After time t realization is revealed, the two pre-
18
dictors are evaluated ex post using MSFE and new fractions of agent types are
19
determined. These updated fractions are used to determine the next period (ag-
20
53
The latest available observation is the best forecast of the future.
53
1
gregate) forecasts of output-and inflation gaps, and so on. Agents’ rationality con-
2
sists therefore in choosing the best-performing predictor using the updated fitness
3
measure. There is a strong empirical motivation for inserting this type of switch-
4
ing mechanism amongst different forecasting rules (see DeGrauwe and Macchiarelli
5
(2015) for a brief discussion of the empirical literature, Frankel and Froot (1990)
6
for a discussion of fundamentalist behaviour, and Roos and Schmidt (2012), Cog-
7
ley (2002), Cogley and Sargent (2007) and Cornea, Hommes and Massaro (2013)
8
for evidence of extrapolative behaviour, in particular for inflation forecasts). More
9
recently, Chiarella et al (2012) performed an empirical validation of a reduced-form
10
heterogeneous agent financial model with Markov chain-regime dependent expecta-
11
tions and showed that such a learning mechanism matches well the boom-bust cycle
12
in the US stock market at the same time as it has good predictability power.
13
14
Just as in Chiarella and Khomin (1999), the aggregate market forecasts of output gap and inflation is obtained as a weighted average of each rule: E˜t πt+1 = αtf E˜tf πt+1 + αte E˜te πt+1
(I.24)
E˜t yt+1 = αtf E˜tf yt+1 + αte E˜te yt+1
(I.25)
15
where αtf is the weighted average of fundamentalists, and αte that of the ex-
16
trapolists. Following Chiarella and Khomin (1999), these shares are time-varying
17
and based on the dynamic predictor selection. The mechanism allows to switch
18
between the two forecasting rules based on MSFE / utility of the two rules, and in-
19
crease (decrease) the weight of one rule over the other at each t. Assuming that the
20
utilities of the two alternative rules have a deterministic and a random component
21
(with a log-normal distribution as in Manski and McFadden (1981) or Anderson
22
et al (1992)), the two weights can be defined based on each period utility for each
23
φ forecast Ui,t , i = (y, π), φ = (f, e) according to:
f απ,t =
f exp(γUπ,t ) f e exp(γUπ,t ) + exp(γUπ,t )
54
(I.26)
f αy,t =
f exp(γUy,t ) f e exp(γUy,t ) + exp(γUy,t )
f e απ,t ≡ 1 − απ,t =
e αy,t
≡1−
f αy,t
=
e exp(γUπ,t ) f e exp(γUπ,t ) + exp(γUπ,t ) e exp(γUy,t ) f e exp(γUy,t ) + exp(γUy,t )
(I.27)
(I.28)
(I.29)
,where the utilities are defined as: f Uπ,t
=−
∞ X
1
f πt−k−1 ]2 wk [πt−k−1 − E˜t−k−2
(I.30)
f wk [yt−k−1 − E˜t−k−2 yt−k−1 ]2
(I.31)
e wk [πt−k−1 − E˜t−k−2 πt−k−1 ]2
(I.32)
e wk [yt−k−1 − E˜t−k−2 yt−k−1 ]2
(I.33)
k=0
f Uy,t
=−
∞ X k=0
e Uπ,t =−
∞ X k=0
e Uy,t
=−
∞ X k=0
and wk = (%k (1 − %)) (with 0 < % < 1) are gemoetrically declining weights
2
adapted to include the degree of forgetfulness in the model (DeGrauwe, 2012). γ is
3
a parameter measuring the extent to which the deterministic component of utility
4
determines actual choice. A value of 0 implies a perfectly stochastic utility. In that
5
case, agents decide to be one type or the other simply by tossing a coin, implying a
6
probability of each type equalizing to 0.5. On the other hand, γ = ∞ imples a fully
7
deterministic utility, and the probability of using the fundamentalist (extrapolative)
8
rule is either 1 or 0. Another way of interpreting γ is in terms of learning from past
9
performance: γ = 0 imples zero willingness to learn, while it increases with the size
10
of the parameter, i.e. 0 < γ < ∞.
11
As mentioned above, agents will subject the performance of rules to a fit mea-
12
sure and choose the one that performs best. In that sense, agents are ’boundedly’
13
rational and learn from their mistakes. More importantly, this discrete choice mech-
14
55
1
anism allows to endogenize the distribution of heterogeneous agents over time with
2
the proportion of each agent using a certain rule (parameter αφ , φ = (f, e)). The ap-
3
proach is consistent with the empirical studies (Cornea et al, 2012) who show that
4
the distribution of heterogeneous agents varies in reaction to economic volatility
5
(Carroll (2003), Mankiw et al (2004)).
6
I.2.2
7
Aggregate Demand:
8
Aggregate Demand:
System of equations
yt = a1 E˜t yt+1 +(1−a1 )yt−1 +a2 (rt −E˜t πt+1 )+(a2 +a3 )ef pt +(a1 −a2 )ψ(ut )kt +Adjt +t (I.34) 9
10
Investment it = e1 E˜t [yt+1 ] + e2 [rt + ef pt − E˜t [πt+1 ]]
(I.35)
11
12
External Finance Premium ef pt = φ¯ nt St
(I.36)
ct = 1 − s t
(I.37)
yt = at (kt ψ(ut ))α hωt 1−α
(I.38)
13
14
Consumption
15
Aggregate Supply:
16
Cobb-Douglas Production Function
17
18
Utilization cost function ψ(ut ) = ξ0 + ξ1 (ut − 1) +
56
ξ2 (ut − 1)2 2
(I.39)
1
Approximated Philips Curve:
2
πt = b1 E˜t πt+1 + (1 − b1 )πt−1 + b2 yt + νt
(I.40)
3
Capital evolution
4
kt = (1 − δ)ψ(ut )kt−1 + Ψit
(I.41) 5
Cash-in-advance constraint
6
St+1 Kt+1 ≤ ϑt [St+1 Kt+1 − Nt ]
(I.42)
7
Labour market
8
yt =
lt wt 1−α
(I.43)
Financial market:
9
Bank net worth
10
¯ St ) nbt = κ(lts + n
(I.44) 11
Evolution of bank leverage
12
τt = τt−1 +
ltd nft
(I.45) 13
Stock market price
14
St =
¯ ] Et [Λt+1 Rts
f [E˜t yt+1 + E˜t πt+1 ] ≡ Rts
(I.46) 15
Firm net worth nft = St n¯t =
16
1 D (L + e1 E˜t yt+1 + e2 (rt + ef pt − E˜t πt+1 )) τ t−1
57
(I.47)
1
2
Deposits dt = dt−1 + st
(I.48)
d ltd = lt−1 + it
(I.49)
ltd = lts
(I.50)
E˜t πt+1 = αtf E˜tf πt+1 + αte E˜te πt+1
(I.51)
E˜t yt+1 = αtf E˜tf yt+1 + αte E˜te yt+1
(I.52)
E˜tf πt+1 = π ∗
(I.53)
E˜tf yt+1 = y ∗
(I.54)
E˜te πt+1 = θπt−1
(I.55)
E˜te yt+1 = θyt−1
(I.56)
3
4
Loan demand
5
6
Credit market equilibrium
7
Learning environment:
8
Inflation learning
9
10
Output learning
11
12
Learning rules:
13
14
15
16
17
Weights f απ,t
=
18
f αy,t
=
f exp(γUπ,t ) f e exp(γUπ,t ) + exp(γUπ,t ) f exp(γUy,t ) f e exp(γUy,t ) + exp(γUy,t )
58
(I.57)
(I.58)
e απ,t
≡1−
f απ,t
=
f e αy,t ≡ 1 − αy,t =
Utilities: f Uπ,t =−
∞ X
e exp(γUπ,t ) f e exp(γUπ,t ) + exp(γUπ,t ) e exp(γUy,t ) f e exp(γUy,t ) + exp(γUy,t )
1
(I.59) 2
(I.60) 3
f πt−k−1 ]2 wk [πt−k−1 − E˜t−k−2
(I.61)
k=0 f Uy,t =−
∞ X
4
f yt−k−1 ]2 wk [yt−k−1 − E˜t−k−2
(I.62)
k=0 e =− Uπ,t
∞ X
5
e wk [πt−k−1 − E˜t−k−2 πt−k−1 ]2
(I.63)
k=0 e Uy,t =−
∞ X
6
e wk [yt−k−1 − E˜t−k−2 yt−k−1 ]2
(I.64)
k=0
Shocks
7
Monetary policy shock:
8
rt = rt−1 + γπt + (1 − γ)yt +
(I.65)
9
Technology shock
10
Yt = At [zt ψ(ut )Kt ]α L1−α
(I.66) 11
Shock to utilization costs
12
ψ(ut ) = ξ0 + ξ1 (ut − 1) +
ξ2 (ut − 1)2 + uct 2
(I.67) 13
uct = ρuc uct−1 + uc
(I.68)
Evolution of stock prices: Just as in De Grauwe and Macchiarelli (2015), the
14
share price is derived from the stable growth Gordon discounted dividiend model:
15
St =
¯ ] Et [Λt+1 Rts
¯ are expected future dividends net of the discount rate, where Λt+1
59
16
1
Rts . Agents in this set-up assume that the 1-period ahead forecast of dividends is a
2
fraction f of the nominal GDP one period ahead, and constant thereafter in t+1,
3
t+2, etc. Since nominal GDP consists of a real and inflation component, agents
4
make forecast of future output gap and inflation according to the specification in
5
subsection 2.3. This forecast is reevaluated in each period. As a result, in order to
6
get the expected (stock) market price, the expected output gap and inflation needs
7
to be defined.
8
I.2.3
9
Following the tradition in this literature, the model is solved using the techniques
10
described in Macchiarelli and De Grauwe (2015), De Grauwe (2012), and described
11
in detail in the online technical appendix.
12
13
Calibration and model solution
To simplify the discussion, we will only present the calibrations of the parameters that are new to this model. A full parameter list can be found in Table I.2.
14
The parameters are set to standard values in the literature. The share of capital
15
in the production α is set to 0.30 as in Boissay et al (2013). Following Christiano et
16
al (2005), Smets and Wouters (2003, 2007) and Gerali et al (2010), we set the capital
17
depreciation rate δ to 0.025. The elasticity of the capital utilization adjustment cost
18
function ψ(ut ) is parametrized to 0.5 as in Smets and Wouters (2007).54
19
The sensitivity of capital (or investment) to changes in the real interest rate e2
20
is, in line with the empirical evidence, set to e2 < 0. To conclude, the parameters of
21
the function determining adjustment costs for capacity utilization (ξ0 , ξ1 , ξ2 ) are set
22
to (0.8, 0.3, 0.25) in order to capture the estimation results of Smets and Wouters
23
(2005) who find that the capital utilization costs are between 0.14 and 0.38 (Euro
24
Area 1983-2002) and 0.21 and 0.42 (US 1983-2002), with a mean of 0.25 (Euro Area)
25
and 0.31 (US). If we normalize ut to 1 (as in Christiano et al (2005), Miao et al (2013)
26
or Auernheimer and Trupkin (2014)), then the cost for utilizing capital will be 0.20
27
(1 − ξ0 ), which is well within the estimated intervals of Smets and Wouters (2005).
28
All shocks, except to the capital utilization, are parametrized as white noise 54 This is equivalent to setting a κi equal to the estimated range (10.18 − 12.81) as in Gerali et al (2010).
60
which means that their autoregressive component is set to 0. Likewise the standard
1
deviations of shocks are set to 0.5 across the entire spectrum.55
2
Table I.2: Parameters of the behavioural model and descriptions Parameter
Description
Value
Calibrated ∗
π d1 e1 d2 d3 e2 a1 0 a1 a2 a3 b1 b2 c1 ψ τ κ e αd n ¯ n ˜ β c2 c3 δ α Ψ κ γ ρ z ξ ξ1 ξ2 z x uc ρk
II
Calibration The central bank’s inflation target Marginal propensity of consumption out of income Coefficient on expected output in investment eq. Coefficient on expected output in consumption eq. to match a1 = 0.5 Coefficient on real rate in consumption eq. Coefficient on real rate in investment eq. to match a2 = −0.5 Coefficient of expected output in output eq. Coefficient of lagged output in output eq. Interest rate elasticity of output demand Coefficient on spread term in output eq. Coefficient of expected inflation in inflation eq. Coefficient of output in inflation eq. Coefficient of inflation in Taylor rule eq. Parameter of firm equity Firms’ leverage Banks’ inverse leverage ratio Equity premium Fraction of nominal GDP forecast in expected future dividends Number of shares in banks’ balance sheets Initial value for number of firms’ shares Bubble convergence parameter Coefficient of output in Taylor equation Interest smoothing parameter in Taylor equation Depreciation rate of capital Share of capital in production Adjustment cost function in investment Adjustment cost in investment parameter Switching parameter in Brock-Hommes (or intensity of choice parameter) Speed of declining weights in memory (mean square errors) Technological development parameter Parameter 1 in the utilization cost function Parameter 2 in the utilization cost function Parameter 3 in the utilization cost function Std. deviation of technology shock Std. deviation of nom. Interest rate shock Std. deviation of financial shock Std. deviation of shock in the utilization cost function AR process of shock to utilization cost function
0 0.5 0.1 0.5 ∗ (1 − d1 ) − e2 −0.01 (−0.5) ∗ (1 − d1 ) − d3 (e1 + d2 )/(1 − d1 ) d2 /(1 − d1 ) (d3 + e2 )/(1 − d1 ) −d3 /(1 − d1 ) 0.5 0.05 1.5 −0.02 1.43 0.09 0.05 0.2 40 60 0.98 0.5 0.5 0.025 0.3 0.5 11 1 0.5 0.5 0.8 0.3 0.25 0.5 0.5 0.5 0.5 0.1
Figures
3
55
The AR-component of the shock to capital utilization cost is set conservatively to 0.1, just enough to generate some persistence in the capital cost structure.
61
62 30
40
50
60
−0.5
20
−0.02
10
0
0
−0.01
60
0.5
50
0
20 30 40 Real interest rate 1
10
0.01
0
1
0
1
60
0.5
1
2
10 20 30 40 50 Entrepreneurial consumption
Output
2
0
0.5
1
1.5
10
10
10
20
30
40
20 30 40 Book value of capital
50
50
20 30 40 50 Entrepreneur net worth
Consumption
Figure II.1: Responses to a productivity shock
60
60
60
10
10
10
20
30
40
20 30 40 Capital stock
20 30 40 Return on capital
Investment
(continued on next page)
0
0.5
1
−0.5
0
0.5
1
0
2
4
50
50
50
60
60
60
63
50
60
30
40
50
60
10
10
10
30 40 Policy rate
20
30
40
20 30 40 Residual earnings
20
TFP parameter
50
50
50
60
60
60
−3
−2
−1
0
0
0.5
1
−0.5
0
0.5
1
10
10
10
50
20
30
40
50
20 30 40 50 Cash−in−advance share
20 30 40 Stock market value
Hours worked
Notes: Impulse responses to an expansionary TFP shock in the DSGE model. The dotted lines represent the responses to the same shock in a canonical financial accelerator (BGG) model.
20
0
10
−0.01
0
60
0.5
50
0.5
20 30 40 Asset price wedge 1
10
0
0.01
1
−0.02
0
0.02
0.04
0
30 40 Inflation
−0.5
20
0.5
0
10
1
Marginal costs
0.5
60
60
60
64
20
30
40
50
60
−2
10
−0.1
0
0
60
0
50 2
20 30 40 Real interest rate
0.1
10
5
5
0
10
10
−0.2
0 60
0
1
10 20 30 40 50 Entrepreneurial consumption
0.2
2
Output
10
10
10
20
30
40
20 30 40 Book value of firm
50
50
20 30 40 50 Entrepreneur net worth
Consumption
60
60
60
10
10
10
20
30
40
20 30 40 Capital stock
20 30 40 Return on capital
Investment
(continued on next page)
0
0.5
1
1.5
−2
0
2
4
0
5
10
15
Figure II.2: Responses to a relaxation in the firm financing costs (external finance premium)
50
50
50
60
60
60
65
60
50
60
10
10
20
30
40
50
20 30 40 50 Cash−in−advance share
10 20 30 40 50 Stock market value of capital
Hours worked
60
60
60
0
1
2
−0.2
0
0.2
10
10
20
30
40
20 30 40 Asset price wedge
Inflation
Notes: Impulse responses to a relaxation (negative shock) of the firm financing costs in the DSGE model. The dotted lines represent the responses to the same shock in a canonical financial accelerator (BGG) model.
−10
40
0 30
−5
1
20
0
2
10
−2
60
−0.02 50
0
0
20 30 40 Residual earnings
2
0.02
10
4
0.04
−2
50
−1 30 40 Policy rate
0
−0.5
20
2
0
10
4
Marginal costs
0.5
50
50
60
60
1
––– Figure II.3: Full impulse responses to an expansionary technology shock with 95% confidence interval
66
67 Figure II.4: Full impulse responses to a relaxation in firm financing conditions with 95% confidence interval
Figure II.5: Evolution of key aggregate variables
68
Figure II.6: Ergodic distributions and learning dynamics
69
0
−18 −20
−5 −22 −24
Magnitude (dB)
Magnitude (dB)
−10
−15
−26 −28 −30
−20 −32 −34 −25 −36
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
−38
500
0
0
−5
−5
−10
−10
−15
−15 Magnitude (dB)
Magnitude (dB)
−30
−20
−25
−30
−30
−35
−35
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
−40
500
20
50
100
150
200
250 300 Frequency (Hz)
350
400
450
500
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
500
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
500
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
500
−20
−25
−40
0
−18 −20
10 −22 −24
Magnitude (dB)
Magnitude (dB)
0
−10
−26 −28 −30
−20 −32 −34 −30 −36
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
−38
500
0
−10
−5
−15
−10
−20
Magnitude (dB)
Magnitude (dB)
−40
−15
−20
−25
−30
−25
−35
−30
−40
−35
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
−45
500
Notes: The following variables are depicted in the figures above (from upper left to right and down): output, investment, capital, consumption, deposits, savings, policy interest rate, and inflation. The dotted lines represented the 95% confidence bands.
Figure II.7: Frequency domain - real variables
70
30
20
20
10
10
Magnitude (dB)
Magnitude (dB)
30
0
−10
0
−10
−20
−20
−30
−30
−40
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
−40
500
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
500
0
50
100
150
200
250 300 Frequency (Hz)
350
400
450
500
−10 −15 −20
Magnitude (dB)
−25 −30 −35 −40 −45 −50 −55 −60
Notes: The following variables are depicted in the figures above (from upper left to right and down): front payment share, utilization costs, loan supply, loand demand, the financing spread, net worth of firms, stock prices and animal spirits. All variables (except the white noise ones) include the 95% confidence bands (dotted red lines).
Figure II.8: Frequency domain - supply and financial variables 71