A real business cycle model with money as a sunspot variable

Journal of Economics and Business 109 (2020) 105891

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A real business cycle model with money as a sunspot variable Matthew S. Wilson

T

Economics Department, Binghamton University, Binghamton, NY 13902 USA

A R T IC LE I N F O

ABS TRA CT

JEL classification: E2 E3 E4 E5

A well-known criticism of the RBC model is that it cannot match the data on money. Due to the perfect flexibility of prices and the absence of frictions, any exogenous increase in the money supply will be fully offset by wage and price increases, implying that money is neutral even in the short run. However, beliefs that money is non-neutral could become a self-fulfilling prophesy. By using money as a sunspot variable in an RBC model, I successfully replicate many of the correlations in the data, even though money does not directly affect the economy’s fundamentals. This shows that models with flexible prices are not necessarily incompatible with the monetary data and offers support for the use of sunspot variables in macroeconomics.

Keywords: Sunspots Indeterminacy RBC Money Increasing returns

1. Introduction If there is indeterminacy, it is possible that the rational expectations solution may include variables unrelated to the fundamentals of the model. Intuitively, it is easy to see how these sunspot variables can matter. If agents believe in the importance of the sunspot variable, they will condition their behavior on it. This change in behavior causes real effects in the economy, thus justifying belief in the sunspot’s significance. However, there is a very natural criticism of sunspot equilibria. If there is a random variable that all agents believe in and whose realization is observed, what specific variable is it? In nearly every model with sunspot variables, this question is left unanswered. This may contribute to the notion described by Benhabib and Farmer (1999) that “indeterminacy is an esoteric area that is unconnected with the core of macroeconomics.”1 One issue that is certainly connected to the core of macroeconomics is the RBC model. The standard version has drawn much criticism for omitting money. If prices truly are perfectly flexible and there are no nominal rigidities, then any increase in the money supply will be fully offset by price increases, so there will be no impact on real GDP. However, this is at odds with much empirical evidence. Many people seem to believe that monetary policy matters, since politicians occasionally mention it and the media reports on the actions of the Federal Reserve. Could it be that we live in a world much like the RBC model, but where money has an impact solely because of people’s self-fulfilling beliefs that it will affect the real economy? To answer this question, I use money as a sunspot variable in an RBC model. Though my model has no nominal rigidities or frictions, it nonetheless matches many of the correlations found in the empirical data, including many facts about money. This shows that an RBC model can explain the evidence on money without relaxing the key assumption that prices are perfectly flexible. It also states a specific and plausible sunspot variable, offering evidence that sunspot variables may be a feature of the real world economy and more than just a theoretical curiosity. The standard RBC model is determinate, but there are several ways to modify it that generate indeterminacy and sunspot

1

E-mail address: [email protected]. Benhabib and Farmer do not endorse this view; they use their chapter in the book to counter it.

https://doi.org/10.1016/j.jeconbus.2020.105891 Received 2 May 2019; Received in revised form 3 January 2020; Accepted 6 January 2020 Available online 16 January 2020 0148-6195/ © 2020 Elsevier Inc. All rights reserved.

Journal of Economics and Business 109 (2020) 105891

M.S. Wilson

solutions. One of the ways is in Guo and Harrison (2001). In a very simple RBC model, they add increasing returns to scale and make the depreciation rate a variable rather than a constant. I follow their approach, though I add government spending, population growth, and labor productivity growth, which is subject to sector-specific shocks. The model pushes the limits of the flexible price framework: in such a setup, is it possible to match the data and provide a causal role for monetary shocks? The standard approach to reconciling the data with an RBC model is to add a banking sector in which productivity shocks also affect money. Thus, monetary shocks would correlate with real GDP but do not cause it. However, this is at odds with a wealth of evidence. With the novel mechanism of money as a sunspot variable, I address this issue. In addition, a sunspot chained to a variable in the real world is more plausible than a sunspot which has no empirical counterpart. This demonstrates a way for indeterminate models to improve their believability and gain broader acceptance. My paper stands at the intersection of the literature on indeterminate RBCs and RBCs with money. Farmer and Guo (1994) was one of the first papers on sunspot RBC models. They used increasing returns to scale in the production function, and this creates multiple equilibria. Suppose that in an equilibrium, agents decided to increase investment and labor. The amount of capital would then increase. For this outcome to be an equilibrium, agents must be behaving optimally, so the decision to invest more in capital must be justified by a rise in the rate of return. Since markets are competitive, the rate of return is the marginal product of capital. If there are increasing returns, the return on capital increases with the level of capital and labor, so agents are not mistaken when they jointly decide to invest more. By the same reasoning, the choice to increase labor is justified by an increase in wages. Thus there are multiple equilibria. The authors claim that their model is at least as good as the standard RBC model at explaining the data. However, their influential paper drew much criticism. Aiyagari (1995) studies the implications for the labor markets. Giving an intuitive rather than mathematical argument, he finds that the standard case of an upward sloping labor supply curve and downward sloping demand curve is not possible in the Farmer-Guo model. The model also required a very high degree of increasing returns in order to create indeterminacy, but the empirical literature suggests that returns to scale are either constant or no more than minimally increasing. One important paper on this subject is Basu and Fernald (1997), which estimates that the aggregate production function is homogenous of degree 1.03. Guo and Harrison (2001) show that in the Farmer-Guo model, indeterminacy can only be achieved if the production function is homogenous of degree 1.49 or more, which is far higher than empirical estimates would allow. For comparison, the standard case of constant returns to scale is a production function that is homogenous of degree one. The main response to these criticisms has been to construct models where the degree of increasing returns is much lower. Benhabib and Farmer (1996) build a two-sector RBC model. One sector produces consumption goods; the other one makes investment goods, which are added to the capital stock. To understand how this helps achieve indeterminacy, revisit the thought experiment from before. The economy starts out in one equilibrium and then agents decide to raise labor and investment. The boost to investment is amplified by the increasing returns in the investment goods sector. This amplification does not occur in a single-sector model. Since the increasing returns are magnified in the two-sector model, the amount of increasing returns can be reduced. Now indeterminacy can be achieved if the production function is homogenous of degree 1.08. This is far closer to constant returns and is empirically plausible. Another advantage of the two sector approach is that the labor supply and demand curves can have their usual slopes, addressing Aiyagari’s critique. Guo and Harrison (2001) refine the two-sector RBC model by adding variable capital utilization. If capital is used more intensely, then production increases, but the depreciation rate rises as well. This gives the agents another way to amplify the increasing returns. Starting from equilibrium, agents may decide to work and invest more. The increasing returns magnify the effects of this decision; if labor and capital both rise by 1 %, then output will rise by more than 1 %. Now suppose that when agents are increasing labor and capital, firms start using capital more intensely. This will boost output even further. As a result, the degree of increasing returns can be reduced and indeterminacy is still achieved. Guo and Harrison (2001) demonstrate this with increasing returns as low as 1.02, which is quite compatible with the empirical literature. I build upon the Guo and Harrison model. Based on the econometric estimates in Burnside (1996), I set the increasing returns to 1.06 and then find that the equilibrium is indeterminate. My model also joins a long tradition of RBCs with money. This literature distinguishes between internal money (transaction services) and external money (created by the central bank). An early paper in this genre was King and Plosser (1984). Though they did not formally solve their model, their approach was extended in papers such as Belongia and Ireland (2006) and Ahmed and Murthy (1994). King and Plosser (1984) added a banking sector that lowered transaction costs; it was modeled as an intermediate good in the production process. Thus, a positive shock to productivity leads to more transactions. As a result, money and output are correlated. However, shocks to external money have no effect on the real economy, since prices are perfectly flexible. Both of us are trying to reconcile the RBC model with the money data. We do so in different ways. They do not allow external money to have a causal role. I do, since in my model, it affects investor confidence and leads to self-fulfilling prophecies. Farmer (1997) finds a way that appears to incorporate money into an RBC model without sacrificing flexible prices and wages or adding any frictions. He does this by putting money in the utility function2 while the rest of the model is a standard RBC. However, giving microeconomic foundations to the MIU framework requires adding a friction such as a cash in advance constraint. My approach avoids any frictions, either explicit or implicit, since I study a limiting case in which an MIU parameter goes to zero.

2 Sossounov (2000) finds mathematical errors in the original paper but shows that the main results can be salvaged if there are increasing returns to scale. Benhabib and Farmer (2000) modify preferences in a different way. They add technical progress directly into the utility function – a very unusual approach – and find that the equilibrium is indeterminate. Another unusual feature is that the labor supply curve has a negative slope; this is important for their results. The negative slope comes from leisure being an inferior good due to real rigidities. Thus, their approach also relies implicitly on frictions.

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Many papers retain the standard RBC assumption that prices are flexible, but then they add frictions. Dia and VanHoose (2017) discuss some of this literature in their survey paper on banking. A cash-in-advance constraint is a popular choice (Barinci & Chéron, 2001; Benhabib & Farmer, 1991; Bosi, Magris, & Venditti, 2005; Christiano & Eichenbaum, 1992). In Bosi, Magris and Venditti (2007), workers face a borrowing constraint. In Freeman and Kydland (2000) and Sustek (2010), using the banking sector’s financial services is costly. Many of these papers feature indeterminacy and sunspots. In contrast to this literature, I avoided adding any frictions. A baseline RBC model is a frictionless economy with flexible prices and wages; those are its key characteristics. To defend that framework, my goal was to make a minimal modification that is enough to reconcile the model with the data. If instead large changes were required, then the model becomes substantially different from a baseline RBC and its success would not vindicate the RBC framework. Increasing returns and sunspots are very modest departures from the usual RBC setup. The production function is now homogenous of degree 1.06 instead of 1.0. The major aspects of the RBC (flexible prices, no frictions) are retained. Another way to minimally alter the RBC framework is to add a tax on nominal capital gains. Some papers have tried this, but they also had to include other frictions. For example, Cooley and Hansen (1989) has a cash-in-advance constraint. In Gavin, Kydland and Pakko (2007), there are transaction costs. Thus, in the previous literature, the researcher is seemingly forced to make a choice. Either sacrifice the assumption that the economy is frictionless or claim that external money is neutral. I find a way to circumvent this dilemma. We can have a frictionless environment and allow external money to impact the real economy. The solution is to make money into a sunspot variable. 2. The model 2.1. Setup of the model There are two sectors, consumption and investment. Let Yct denote the amount of consumption goods produced at time t. The production function is

Yct = Bct (ut K ct )α (Act Lct )1 − α

(1)

The term ut denotes capital utilization. As capital is used more intensely, ut increases and production rises. K ct and Lct are capital and labor in the consumption sector, respectively. The term Bct is an aggregate production externality, which is defined as

Bct = [(u¯ t K¯ ct )α (Act L¯ ct )1 − α ]θ

(2)

Here the bars represent economy-wide averages. Since all firms are identical, in equilibrium ut = u¯ t , K ct = K¯ ct , and Lct = L¯ ct , so the aggregate production function for consumption goods will be homogenous of degree (1 + θ) in capital and labor. The standard assumption of constant returns to scale corresponds to θ = 0 . The investment goods sector is symmetric:

YIt = BIt (ut KIt )α (AIt LIt )1 − α

(3)

BIt = [(u¯ t K¯ It )α (AIt L¯It )1 − α ]θ

(4)

Labor productivity improves exponentially at rate gA in both sectors:

∼ logAct = A¯ + gA t + Act

(5)

∼ logAIt = A¯ + gA t + AIt

(6)

∼ ∼ The labor productivity shocks, Act and AIt , are AR(1) but can be different for each sector. Both εac, t and εai, t are mean zero, white noise shocks. ∼ ∼ Act = ρac Ac, t − 1 + εac, t

(7)

∼ ∼ AIt = ρai AI , t − 1 + εai, t

(8)

The depreciation rate δt depends on how intensely capital is used.

δt =

utγ , γ>0 γ

(9)

From the production functions it is clear that as capital is used more intensely, production increases. The tradeoff is that depreciation also rises. The capital accumulation equation is

Kt + 1 = (1 − δt ) Kt + It

(10)

In contrast to a one sector RBC, It is the number of investment goods demanded and is not the amount of savings. This is important for achieving indeterminacy, since the production of investment goods serves to amplify the increasing returns. There are H identical households; their utility function is given below. 3

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⎡ ⎛ 1−σ ⎢ ⎛ l jt ⎞ + ⎜ U = ∑ β t Nt H−1 ⎢logcjt − ⎜ z ⎟ ⎜⎜ t=0 ⎢ ⎝1 − σ ⎠ ⎝ ⎣ ∞

mjt 1 −ζ

( ) pct

− 1⎞ ⎤ ⎟ fˆ ⎥, σ ≤ 0, z > 0 ⎥ 1−ζ ⎟⎟ ⎥ ⎠ ⎦

(11)

Here cjt and l jt are consumption and labor per person in household j. Variables denoted with uppercase letters are aggregates; per capita variables are lowercase. Nominal money per person is mjt ; the price of one unit of consumption is pct . Nt is population at time t. Thus, the household’s utility depends on consumption, real money, and labor per person. There are Nt / H people in each household. Population grows exponentially at an exogenous rate n.

logNt = N¯ + nt

(12)

Let pIt be the price of investment goods. The relative price ratio pt equals pIt / pct . The real GDP equation is below.

Yt = Ct + pt It + Gt

(13)

The government collects lump-sum taxes Xt and maintains a balanced budget. Its spending Gt grows exponentially and is subject to AR(1) shocks. The term εGt is a white noise random variable.

Xt = Gt

(14)

∼ logGt = G¯ + (n + ϕ) t + Gt

(15)

∼ ∼ Gt = ρG Gt − 1 + εGt

(16)

Here ϕ is the per capita growth rate of GDP on the balanced growth path. Since there are increasing returns to scale, this rate will differ from the growth rate of labor productivity gA . Since there are two sectors, total capital Kt is capital in the consumption sector plus capital in the investment sector. Total labor is similar.

K ct + KIt = Kt

(17)

Lct + LIt = Lt

(18)

All the firms are identical, so in equilibrium capital, labor, and capital utilization will be the equal across firms.

ut = u¯ t

(19)

K ct = K¯ ct

(20)

KIt = K¯ It

(21)

Lct = L¯ ct

(22)

LIt = L¯It

(23)

The government purchases consumption goods, and in equilibrium supply equals demand.

Yct = Ct + Gt

(24)

YIt = It

(25)

The household maximizes its utility function (11) subject to the constraints given below.

N ΔMt pct cjt + pct pt i jt + pct x t + mjt = ⎛ t − 1 ⎞ mj, t − 1 + pct rt kjt + pct wt l jt + N Nt t ⎝ ⎠

(26)

N kj, t + 1 = ⎛ t − 1 ⎞ ((1 − δjt ) kjt + i jt ) ⎝ Nt ⎠

(27)

⎜

⎜

⎟

⎟

Eq. (26) is the household’s budget. On the left hand side, we see that the household spends its resources on consumption, investment goods, taxes, and holding money. Recall that mj, t − 1 is money per person; since the household has grown, money per person from the previous period is

( )m Nt − 1 Nt

t − 1.

For example, if the household doubles, then money per person is cut in half (the same

adjustment also appears in the capital accumulation equation). In the budget constraint (26), the household earns the rental rate rt on its capital and wages wt from its labor. The last term is an injection of new money created; the aggregate money supply is discussed in Section V. Unlike many other RBCs with money, there is no banking sector in this model. Thus, in the rest of the paper, “money” will always refer to “external money.” 2.2. First order conditions Now we can start solving the model. Both households and firms are price takers, so the first order conditions for the firm are 4

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straightforward. Firms rent capital and labor until their marginal products equal the rental rate and the wage, respectively. Clearly the marginal products must be equal across sectors, or else households will reallocate their resources toward the sector with the higher wage or interest rate.

rt =

wt =

αp YIt αYct = t = MPK K ct KIt

(28)

(1 − α ) pt YIt (1 − α ) Yct = = MPL Lct LIt

(29)

The first order condition for labor is the following.

zct = wt ltσ

(30)

In equilibrium, all households are identical, so l jt = lt for all households j. The first order condition for investment can be combined with the envelope condition to produce the investment Euler equation.

rt + 1 + pt + 1 (1 − δt + 1 ) ⎤ β 1 = Et ⎡ ⎥ ct pt ⎢ ct + 1 ⎣ ⎦

(31)

Households decide on how much real money to hold subject to their budget constraint. They treat in a money Euler equation.

Δ Mt Nt

as exogenous. This results

−ζ

pct 1 1 ⎤ ⎛ mt ⎞ ˆ ∙ = βEt ⎡ ⎥ + ⎜p ⎟ f ⎢p ct c t +1 ⎦ c , t 1 + ⎝ ct ⎠ ⎣

(32)

It has an intuitive interpretation. Agents can either consume or hold money, which is then used for consumption in the next period. In order to optimize, the marginal utility of consuming today must be equal to the marginal utility of holding money plus the expected marginal utility of consuming next period. Consumption in the next period is discounted by β and adjusted for changes in the price level. The transversality condition is

lim Et β t + τ

τ →∞

kt + τ + 1 =0 ct + τ

(33)

Optimal investment is derived by inserting optimal consumption, money, and labor into the household’s budget constraint. A final first order condition is necessary to determine capital utilization ut . Think of ut being set by negotiations between households and firms. Due to perfect competition, neither households nor firms have any market power, so ut is set so that the marginal gain to firms (increased production) equals the marginal loss to households (faster depreciation of capital)3 . Guo and Harrison (2001) use this method but do not prove that it is valid; a proof is in Appendix C.

α (Yct + pt YIt ) = pt utγ − 1 Kt ut

(34)

The presence of capital utilization lowers the degree of increasing returns necessary for indeterminacy. Suppose that from an equilibrium, capital and labor are increased by the same proportion. Then output rises by a larger proportion due to the increasing returns. If Eq. (34) is solved for ut , then it can be seen that ut will also rise. This is because output increases more than capital does. Since production depends positively on ut , output is boosted further. The increasing returns are thus amplified, so they can be decreased and still produce indeterminacy. 2.3. Balanced growth path To further analyze the model, it is convenient to define de-trended versions of the variables.

ct = ∼ ct e ϕt

(35)

∼ it = it e ϕt

(36)

∼ kct = kct e ϕt

(37)

∼ kIt = kIt e ϕt

(38)

∼ e ϕt wt = w t

(39)

3 Of course, it would be possible to allow capital utilization to vary across sectors if the rental rate also varied. However, this would make the model much more complicated while it is not clear if this would yield any additional insights.

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∼ kt = kt e ϕt

(40)

gt = ∼ gt e ϕt

(41)

∼ mt = (e ϕm )t m t

(42)

pct = (e ϕp )t∼ pct

(43)

Now we search for the balanced growth path. The production functions and the money Euler equation can be rewritten as follows. ∼ ∼ ∼ ct + ∼ gt = ((ut kct )α (lct )1 − α )1 + θe (N¯ + nt ) θ + (1 + θ)(1 − α )(g A t + A¯ + Act ) + (α (1 + θ) − 1) ϕt (44) ∼ ∼ ∼ it = ((ut kIt )α (lIt )1 − α )1 + θe (N¯ + nt ) θ + (1 + θ)(1 − α )(g A t + A¯ + AIt ) + (α (1 + θ) − 1) ϕt

(45)

∼ ∼ −ζ pct e−ϕp 1 ⎤ eϕ m ∙ ∼ ⎥ + e ϕfˆ et (ϕ + Δ ϕp − Δ ϕm) ⎜⎛ ∼t ⎞⎟ = βEt ⎡ ∼ ∼ ⎢ ct ⎝ pct ⎠ ⎣ pc, t + 1 ct + 1 ⎦

(46)

After removing the trend, there should not be any more exponential growth. Therefore:

nθ + (1 + θ)(1 − α ) gA + (α (1 + θ) − 1) ϕ = 0

(47)

ϕ + ζϕp − ζϕm = 0

(48)

This imposes restrictions on the parameters ϕ and ζ .

ϕ=

nθ + gA (1 − α )(1 + θ) 1 − α (1 + θ)

(49)

ζ=

ϕ ϕm − ϕp

(50)

If there were constant returns to scale, then θ = 0 . This would lead to the usual result ϕ = gA ; i.e., the trend growth rate is equal to growth rate of productivity. However, the presence of increasing returns alters this story. For instance, if population growth increased (n↑) , then the growth rate of GDP per capita would also increase. This is because increasing the inputs by 1 % boosts production by more than 1 %. 3. Calibration I use quarterly US data from 1983–2007. The sample period was chosen so that there would be no structural breaks in the money regressions (see Section V). I set the discount factor β = 0.99. The population parameters N¯ and n are computed directly from US Census data. The quarterly per capita growth rate on the balanced growth path, ϕ , is set at 0.005. In the large majority of macro models, quarterly depreciation is calibrated to 2.5 %. I choose the depreciation parameter γ so that the model’s steady state depreciation rate is the same.

δ=

α (c + g + i p) = .025 γpk

(51)

Absence of time subscripts denotes steady state, since the steady state is independent of time. On average, government spending is about 20 % of GDP, so I calibrate G¯ accordingly.

g = .2 c + g + ip

(52) 4

I set α = 0.25. Most calibrations choose a higher α , but that has implications that are not very compatible with the data . The steady state investment to GDP ratio is

ip α (e n + ϕ − 1 + δ ) = c + g + ip γδ

(53)

On average, the investment to GDP ratio is between 0.16 and 0.17 in the data. If α is .25, then the steady state ratio is about 20 %, while a typical choice of α = .33 yields a steady state investment ratio of nearly 27 %. There is no support for this in the empirical data. It would be interesting to check other RBC calibrations with a higher α and see if their investment to GDP ratio is empirically plausible. Microeconomic studies commonly find that households use about 30 % of their time for labor. I select the labor-leisure parameter z accordingly. 4 Caselli and Feyrer (2007) estimate the MPK and correct for the presence of non-reproducible capital. This leads to estimates of α that are lower and much closer to 0.25 than to the usual 0.33.

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Table 1 Calibration values summary. Variable

Value

Description

n N¯ β γ G¯

Population parameter Population parameter Quarterly discount rate Depreciation parameter Government spending parameter

z σ θ ρac ρG ϕm ϕp

.002665 log(203,302,031) 0.99 1.6066 17.6699 5.6648 −0.2503 0.06 0.95 0.95 0.0094 0.0074

ζ α ϕ m15 f^

2.4817 0.25 0.005 8.3931 < 0.0001

MIU parameter Reproducible capital share Quarterly real GDP growth parameter Money sunspot coefficient MIU parameter

Labor-leisure parameter Labor-leisure parameter Increasing returns parameter AR(1) coefficient for consumption sector shocks AR(1) coefficient for government spending Money growth parameter Inflation parameter

1

(1 − α )(c + g + i p) ⎞ 1 − σ l=⎛ = .3 zc ⎝ ⎠

(54)

The labor productivity constant A¯ has no effect on the dynamics and is normalized to 0. The increasing returns parameter θ is .06, based on Burnside (1996). The labor productivity growth rate parameter gA was determined by my choice of ϕ . The number of households (H) has no impact on the model since it drops out in the first order conditions. The AR(1) coefficients ρac and ρG are both set to 0.95. Because the intensity of capital utilization is a variable rather than a constant, identifying the labor productivity shocks is more difficult than usual. For more discussion on why ρac is 0.95, see Appendix A. Now I can estimate the standard deviation of government shocks and consumption sector labor productivity shocks, εGt and εac, t , respectively. Based on the money and price growth rates that appear in the data for 1983–2007, I calibrate ϕm = 0.0094 and ϕp = 0.0074 . This implies ζ = 2.4817. The parameter fˆ can be set arbitrarily close to zero so that the model collapses to a pure RBC. The two remaining parameters, σ and m15 , were set so that the equilibrium is indeterminate and that the model matches the standard deviation of M2 (Table 1). 4. Dynamics

ct / c ) . There are two non-predetermined variHere the variables are rewritten as log deviations from steady state, e.g. cˆt = log(∼ ables, investment and the price of consumption goods.

ˆ t + m5 gˆ ˆ t + m4 ai Et [iˆt + 1] = m1 iˆt + m2 cˆt + m3 ac t

(55)

ˆ t + 1 + m10 cˆt + m11 iˆt + m12 gˆ + m13 ac ˆt ˆ t + 1 + m9 ai ˆ t + m14 ai cˆt + 1 = m6 iˆt + 1 + m7 gˆt + 1 + m8 ac t

(56)

ˆ t − 1 + m25 iˆt + m26 gˆ + m27 ac ˆt ˆ t + (1 − Λ) pˆct + m20 cˆt − 1 + m21 iˆt − 1 + m22 gˆt − 1 + m23 ac ˆ t − 1 + m24 ai ˆ t + m28 ai Et [pˆc, t + 1 ] = Λm t

(57)

Let ξit and ξpc, t denote the forecast errors for investment and the price of consumption goods, respectively. The bold terms are vectors that may contain up to ℓ lags. This is necessary because lags show up after some variables are substituted out; this can be seen in the equations above.

⎛ ⎜ ⎜ ⎜ ⎜ μt ≡ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎛ aˆ c, t − 1 ⎞ ⎞ ⎜ aˆ I , t − 1 ⎟ ⎟ ⎟ ⎜ gˆ ⎟ t−1 ⎟ ⎜ ˆ ⎟ ε ⎛ ac, t ⎞ ⎜ it − 2 ⎟ ⎟ ξ ε cˆt − 1 ⎟ = Γ1 ⎜ cˆt − 2 ⎟ + φ ⎜ aI , t ⎟ + Π ⎛ pc, t ⎞ ⎜ξ ⎟ ε gt ⎟ ⎜ pˆc, t − 1 ⎟ ⎜ pˆc, t − 2 ⎟ ⎝ it ⎠ ⎝ εmt ⎠ ⎟ ⎜ ⎟ ˆt m ˆ t−1 m ⎟ ⎜ ⎟ pˆct pˆc, t − 1 ⎟ ⎜ ⎟ ⎜ iˆt − 1 ⎟ iˆt ⎟ ⎠ ⎠ ⎝ aˆ ct aˆ It gˆt ˆit − 1

(58)

I solved the model using the method of Lubik and Schorfheide (2003). Using the generalized Schur decomposition, I rewrite the 7

Journal of Economics and Business 109 (2020) 105891

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system as follows.

′ (Q1

ω11 Q2′ ) ⎛ 0 ⎝

ε ⎛ ac, t ⎞ εaI , t ω12 ′ Ω11 Ω12 ⎞ ′ ⎛ ξpc, t ⎞ ′ ′ ⎛ ⎜ ⎞Z μ = Zμ + φ ε ⎟ + Π⎜ ⎟ ω22 ⎠ t (Q1 Q2 ) ⎝ 0 Ω22 ⎠ t − 1 gt ⎟ ⎜ ⎝ ξit ⎠ ⎝ εmt ⎠ ⎜

⎟

(59)

Since there is only one explosive root, the dimensions of ω22 and Ω22 are 1 × 1. Thus, the stability condition will generate just one equation. This is not enough to uniquely solve for the two unknowns, ξpc, t and ξit . That is why the equilibrium is indeterminate and sunspot solutions exist. Because QQ′ = I , the system can be simplified further.

ε ⎛ ac, t ⎞ ξ ε ω11 ω12 ′ Q Ω Ω 1 11 12 ′ ⎞ Z μ + ⎛ ⎞ φ ⎜ aI , t ⎟ + ⎛ Q1 ⎞ Π ⎛ pc, t ⎞ ⎛ ⎞ Z μt = ⎛ t−1 ⎜ ⎟ ε ⎝ 0 ω22 ⎠ ⎝ 0 Ω22 ⎠ ⎝Q2 ⎠ ⎜ gt ⎟ ⎝Q2 ⎠ ⎝ ξit ⎠ ε mt ⎠ ⎝ ⎜

⎟

⎜

⎟

⎜

⎟

(60)

Now I apply the singular value decomposition to Q2 Π . ′

D 0 ⎛V ⎞ Q2 Π = (U1 U2 ) ⎛ 11 ⎞ ⎜ 1′ ⎟ ⎝ 0 0 ⎠ ⎝V 2 ⎠

(61)

According to the method of Lubik and Schorfheide (2003), the full set of stable solutions is

ε ⎛ ac, t ⎞ ε ⎛ ξpc, t ⎞ ′ −1 ⎜ aI , t ⎟ ⎜ ξ ⎟ = (V2 M1 − V1 D11 U1 Q2 φ) εgt + V2 M2 St ⎟ ⎜ it ⎝ ⎠ ⎝ εmt ⎠

(62)

Here St is the sunspot; M1 and M2 are matrices of coefficients that the researcher can freely choose. The number of rows in the M1 and M2 matrices are (number of nonpredetermined variables ) − (number of explosive roots ) . Therefore, when the system is determinate, the M1 and M2 matrices drop out; there is a unique solution, so the researcher cannot pick any of the coefficients. In this model, the dimensions of M1 and M2 are 1 × 4 and 1 × 1, respectively. I chose the simplest nontrivial case. I selected the coefficients of M1 and M2 so that the investment forecast error depended solely upon the money sunspot. I.e., a suitable choice of M1 made the bottom row of −1 (V2 M1 − V1 D11 U1′ Q2 φ) into a 1 × 4 vector of zeros.

ξit = m15 St , m15 > 0

(63)

This formalizes the story told earlier in the Introduction. Money does not affect the real economy’s fundamentals. However, agents believe that it does, so it influences their behavior. Specifically, investment rises after a money sunspot shock. It is well known that stock markets rally after the Fed cuts rates and increases the money supply. Thus, the money sunspot shock may be interpreted as a boost to investor confidence. It causes investment to be higher than previously anticipated, so that is why money appears in the forecast error. Since there are increasing returns to scale, a rise in investment can lead to a rise in the rate of return. Thus, the increase in investor confidence is a self-fulfilling prophesy. This also affects the price of consumption goods. Intuitively, when the money supply grows faster than expected (St > 0) , prices also rise more than expected. The next section discusses the money sunspot in more depth. 5. The money sunspot For St to be a valid sunspot, it must satisfy the condition Et − 1 St = 0 , so it is more accurate to say that St represents unanticipated money. Anticipated money cannot play any role since its conditional mean is nonzero. There are several methods for decomposing monetary policy into its anticipated and unanticipated components. Kuttner (2001) analyzed the futures market for the federal funds rate. He focuses on the one day change in the spot month contracts; the change that occurs after an FOMC meeting is the unanticipated shock. Romer and Romer (2004) took a different approach. They studied the Fed’s “Greenbook” forecasts. These estimates are an important input into the FOMC’s decision. Romer and Romer (2004) run regressions to determine how the FOMC changes the federal funds rate in response to the forecasts. The residuals are the unanticipated shocks. An older literature ran regressions on M2 growth, defining the fitted values to be anticipated money and the residuals to be unanticipated money. These two components were then used in regressions on GDP, unemployment, or other variables of interest. This literature died out in the late 1980s and early 1990s without a clear consensus. The conclusions were sensitive to the number of lags and choice of independent variables. Barro (1977), Barro (1978) and Fackler and Parker (1990) found that anticipated money had no impact, while Frydman and Rappoport (1987), McGee and Stasiak (1985), and Mishkin (1982) reached the opposite conclusion. However, the approaches of Kuttner (2001) and Romer and Romer (2004) were unsuitable for the sunspot. I ran several tests to check if the conditional mean zero assumption was satisfied. Kuttner (2001) and Romer and Romer (2004) didn’t survive the very first test; both of them had autocorrelation. Thus, a rational agent would be able to anticipate these “unanticipated” shocks. Kuttner’s (2001) method is premised on the efficiency of futures market. However, if his approach is correct, then the autocorrelation in his 8

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series implies that a trader could grab a “free lunch” in the futures market. This would refute the assumption that his method based on. As a result, even if the autocorrelation were extracted from his series and I just took the innovations (i.e., regressing yt = ρyt − 1 + ωt and using ωt ), I cannot be sure that those truly represent unanticipated shocks. For the Romer and Romer (2004) series, I suspected that the autocorrelation was due to a structural break in how the Fed responded to the Greenbook forecasts. There was a structural break in the autocorrelation; with both of those to worry about, I feared that the series would be unsalvageable for use as a sunspot variable. Thus, I went with the approach in the older literature. I run regressions on the de-trended log of M2 growth and consider the residuals to be unanticipated money. These residuals will be the sunspot variable. If the log-linearized equations of the model are used, then all the endogenous variables can be written as functions of the exogenous shocks. Thus, predicting M2 growth using the variables in the agents’ information set will almost surely involve lags of the exogenous shocks. Lags of M2 growth, GDP, and the CPI may also be important. Of course, the unemployment rate could also matter, but it is not in the agents’ information set in the model; there is no unemployment in RBC models. However, labor is in the model. Its empirical counterpart is total hours worked. Government shocks are calculated by plugging the real government spending data into Eqs. (15) and (16) of the model. Labor productivity shocks are computed using the Solow residual; the details are in Appendix A. I regressed the de-trended log of M2 growth on six lags of itself, government shocks, the Solow residual, labor, GDP, and the core CPI. The last three variables in the list were HP filtered. The sample period was 1968 – 2007. 5

log(Mt + 1) − log(Mt ) − ϕm =

∑ Li [βi (log(Mt ) − log(Mt−1) − ϕm) + αi yt

∼ ∼ + γi CoreCPIt + χi Act + ηi Gt + κi lt ] + εm, t + 1

i=0

(64)

Additional lags were insignificant. I checked for whether there was a structural break in the early 1980s. We would expect monetary policy to change after Paul Volcker became the Fed chair. There is very strong evidence of a structural break between 1982Q4 and 1983Q1 (p-value < 0.001 from an Andrews (1993) test). I do not attempt to study the periods before 1983 or after 2007. Before 1983, there is likely another structural break around the time that the Bretton Woods convention was unraveling. However, that would leave me with about 40 observations when there are more than 30 variables in the regression. Hardly any degrees of freedom would be left. A similar problem exists for the data after 2007: too many explanatory variables and too few observations. The period after 2007 is certainly of great interest; as more data comes in, I may explore this in future work. In this paper, the sample period is 1983–2007. Surprisingly, labor was insignificant, so it was dropped from the original regression. Thus, my results are based on the following. 5

log(Mt + 1) − log(Mt ) − ϕm =

∑ Li [βi (log(Mt ) − log(Mt−1) − ϕm) + αi yt

∼ ∼ + γi CoreCPIt + χi Act + ηi Gt ] + εm, t + 1

i=0

(65)

I model the Fed as setting money according to Eq. (65) for the period 1983–2007. Of course, we know the Fed targets the interest rate rather than the money supply. However, changing the money supply is the tool it uses in order to hit its interest rate target. In principle, it should be possible to have an interest rate sunspot instead of a money sunspot. The sunspot can be any conditional mean zero random variable. Thus, it is entirely possible to have a money sunspot even though the Fed is targeting the interest rate. Future research may explore interest rate sunspots, but as discussed earlier, I was unsuccessful in finding a series that satisfied the conditional mean zero requirement. The residuals from the regression in Eq. (65) represent unanticipated money and will be the sunspot St in the model. A necessary condition is that Et − 1 St = 0 , so it must be the case that no variables in the model can predict the residual from the regression. The first test checks for autocorrelation in the residual, which would imply that the sunspot would be able to predict future values of itself. The outcomes are presented in Table 2. There is no evidence of autocorrelation. Next, I regressed the residuals on lags of other variables in the model, including lags of the sunspot. Kuttner (2001) and Romer and Romer (2004) construct their measures of anticipated monetary policy with variables that should be in the agents’ information sets. Earlier I explained why their unanticipated shocks were unsuitable. However, if my sunspot truly has a conditional mean of zero, then no known variable should be able to forecast it – even variables that mis-measure anticipated policy. Consistently, there is no evidence that any of the variables can predict the sunspot. Thus, there is no evidence that the assumption Et − 1 εmt = 0 is violated. See Appendix B for the results of these regressions. In the simulations, the residuals from the regression are used as the sunspot variable (St = εmt ). The idea is to use unanticipated money as a sunspot variable, rather than a sunspot variable whose variance is calibrated to match the variance of unanticipated money. This is because the sunspot should correspond to some variable in the real world. The notion that everyone observes and conditions their behavior on a random variable does not seem plausible if we cannot specify what the random variable is. Using the residuals from the regression connects the sunspot to real world data, and thus can support the case that sunspots may exist in reality Table 2 Tests for autocorrelation in the residuals. Variable

Coefficient

Std. error

t stat

p-value

residt-1 constant

0.0182 0.0001

0.1003 0.0004

0.18 0.36

0.856 0.718

9

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Fig. 1. Realizations of the sunspot variable 1983–2007.

and not just in theory. Fig. 1 shows the time series plot for the sunspot. In the simulations, I generated the sunspot by taking random draws with replacement from the distribution of residuals during the sample period 1983–2007. Now that all the shocks have been defined, Table 3 summarizes their properties. I compared the sunspot to other measures of monetary policy shocks. Since they are measuring the same thing, I expected them to be correlated. First, I ran regressions with Romers’ federal funds rate shock. As expected, the correlation is negative; an increase in the federal funds rate corresponds to a decrease in the money supply. Table 4 shows the results. The coefficient on the sunspot narrowly fails to reach the 5 % threshold of significance. However, that changes if I add one lag of the sunspot. This is demonstrated in Table 5. Further lags were insignificant. The sunspot was uncorrelated with the shocks in Kuttner (2001). However, there was very little overlap in the time horizons; only 19 observations from Kuttner (2001) could be paired with an observation of the sunspot. With Kuttner’s methodology, there is no shock if the FOMC does not hold a meeting. In such a small sample size, it is difficult to clear the threshold for statistical significance. Overall, the money sunspot variable satisfies the conditional mean zero condition and is correlated with another measure of monetary policy shocks. 6. Impulse response functions Fig. 2 shows how the economy responds to labor productivity shocks in the consumption sector. The units are proportional deviations from steady state, e.g. 0.1 is 10 % above steady state. One interpretation of the graph is that if the model is in steady state in period 0, and labor productivity is 1 % above steady state in period 1, then GDP will be about 0.65 % above steady state in period 1. Since the labor productivity shock directly impacts the consumption sector while only indirectly affecting investment, the consumption IRF is quite responsive relative to investment and GDP. For reasons discussed in Appendix A, investment sector labor productivity shocks are set to zero. Fig. 3 shows the IRFs for the CPI and M2. There is some intuition for the results. Initially, output rises; all else equal, this causes the price level to fall since supply has increased. However, in the future the Fed responds by increasing the money supply. As a result, prices start to rise again. It is less clear why M2 falls initially. The Fed could reduce the volatility of prices by raising M2 immediately after a labor productivity shock occurs. In the money regression (Eq. 65), there was a negative coefficient on the first lag of labor productivity shocks, but it was not statistically significant. Perhaps there is a lag between when a shock happens and when it is observed by agents – including the Fed. Future work may explore this. The model is not very sensitive to government spending shocks, as illustrated in Fig. 4. This IRF is more compatible with the empirical result that investment is more volatile than GDP which is more volatile than consumption. The reason for the decaying oscillations in these graphs is that the eigenvalues are complex. This differs from most theoretical models but it is quite compatible with a number of empirical studies. Azariadis, Bullard and Ohanian (2004) estimate several VAR and AR models using a variety of de-trending methods. They frequently find evidence of complex eigenvalues. The IRFs in Blanchard and Quah (1989) and Nelson and Plosser (1982) also exhibit decaying oscillations, suggesting the presence of complex eigenvalues. Table 3 Summary statistics for shocks. Variable

Std. dev.

εmt εGt εac, t

0.0038 0.0087 0.0065

10

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Table 4 Regression output. The dependent variable is the federal funds rate shock in Romer and Romer (2004). Variable

Coefficient

Std. error

t stat

p-value

Sunspott constant

−17.453 0.0792

8.9795 0.0360

−1.94 2.20

0.057 0.032

Table 5 Regression output. The dependent variable is the federal funds rate shock in Romer and Romer (2004). Variable

Coefficient

Std. error

t stat

p-value

Sunspott Sunspott-1 constant

−22.490 −10.629 0.0628

8.3886 8.2867 0.0332

−2.68 −1.28 1.89

0.010 0.205 0.064

Fig. 2. Impulse response functions for a consumption sector labor productivity shock.

Fig. 3. Impulse response functions for a consumption sector labor productivity shock.

Fig. 4. Impulse response functions for a government spending shock.

11

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Fig. 5. Impulse response functions for a government spending shock.

Fig. 5 shows that neither money nor prices is very responsive to government spending shocks. Recall that the CPI corresponds to pct in the model, not pt ( pt is the price of investment goods relative to consumption goods). Fig. 6 shows the IRF for the money sunspot. From Eq. (63), we know that agents react to the sunspot shock by investing more. Effectively, this unexpected increase in the money supply is boosting investor confidence. To finance the higher level of investment, they must decrease consumption and increase labor. Otherwise they would be violating their budget constraint. Is it rational for agents to respond to the sunspot in this way? Due to the increasing returns, the wage and rental rate rise when labor and investment rise. Thus, there is nothing irrational about the agents’ choices to work and invest more when there is a money sunspot shock. They believe that a money sunspot shock will lead to higher wages and rental rates, and this belief is a self-fulfilling prophesy when they increase labor and investment. Fig. 7 shows the IRFs for the CPI and M2. By definition, a 1 % shock to the money sunspot means that M2 is higher than anticipated, so M2 rises in the first period. Prices increase in order to offset the higher money supply. However, prices do not completely adjust in the first period. This is not due to any frictions or rigidities. It is because the sunspot boosts investor confidence. They believe that the rate of return on investment will rise, so they invest more and it becomes a self-fulfilling prophecy. Since investment has risen, GDP increases as well. In the second period, the Fed reduces the money supply in order to fight inflation. However, the Fed has a competing goal; it also desires to stabilize output. GDP’s strongest response to the sunspot occurs in the first period. After that, it immediately begins to fall, though it remains above steady state for several periods. To combat this fall in GDP, the Fed starts increasing the money supply with a lag. Eventually, GDP stops declining, though this is not because the Fed’s efforts to help. Anticipated money is still neutral. GDP stops falling and starts to rise because the eigenvalues are complex; this implies that the dynamics will exhibit oscillations. Once GDP rises again, the Fed reacts by cutting the money supply. The oscillations continue: later, GDP falls so the Fed boosts M2 again, and so on. In the model, optimal monetary policy is very straightforward. Since the money sunspot has a conditional mean of zero, all that the Fed can do is add noise. If, for instance, the Fed fights recessions by increasing the money supply, then rational agents will have anticipated it. At best, it has no impact (remember that anticipated money is neutral). At worst, people may underestimate or overestimate the Fed’s response, so the shocks end up increasing the volatility in the economy. To eliminate the noise, the Fed should target M2. The exact rule for setting the target does not matter; as long as there is some announced target, then the unanticipated shocks vanish and the economy becomes more stable. In a more sophisticated model, both the sunspot and anticipated money would matter. Future work may examine this, but for the purposes of illustrating the monetary sunspot concept, a more parsimonious framework is preferable. Future research may also study the relationship between the sunspot shock in the interest rate. The current model cannot do that because there are no bonds in my simple setup.

Fig. 6. Impulse response functions for a money sunspot shock. 12

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Fig. 7. Impulse response functions for a money sunspot shock.

7. Simulation results After starting from steady state, I ran 1000 simulations of 100 periods each, corresponding to the number of quarters from 1983–2007. Table 6 compares the standard deviations in the simulations to the data. As usual in the RBC literature, I applied the HP filter. However, this is a controversial technique, so I also report results for the Hamilton filter. According to Hamilton (2018), it fixes many of the issues with the HP filter. The standard deviations in the model are close to their empirical counterparts. The model successfully reproduces the stylized fact SD (I ) > SD (Y ) > SD (C ) . Since money is used as a sunspot variable, a key test of the model is whether it can match the evidence on money. Because the sunspot must have a conditional mean of zero, only unanticipated money can cause changes in output. However, anticipated money can still correlate with output, since the variables that cause agents to anticipate money can also affect GDP. By plugging the simulated data into the regression results for anticipated money, I can construct expected money in the model. Adding these fitted values to the sunspot, which was formed from the residuals, yields the M2 growth rate that would occur in the model. Table 7 compares results between the model and the data from 1983–2007. With a few exceptions, the correlations in the model are quite similar to the ones that appear in the data. Table 8 shows the correlations between M2 and lags of GDP. The model does not do as well with these correlations, though it still matches many of them. Figs. 8 and 9 are graphs of the two previous tables. The points are the correlations in the data; the shaded region is the 95 % confidence interval in the model. As an example, consider the point that is furthest to the left in Fig. 8. It corresponds to the correlation between Mt and Yt − 8. In the data, this correlation is 0.3377. In 95 % of the model’s simulations, corr (Mt , Yt − 8) was between 0.15 and 0.4, so that area is shaded. In the large majority of cases, the correlations in the data lie within the model’s 95 % confidence interval. The model is also successful at replicating the autocorrelation in GDP. A few of the eigenvalues are close to one in absolute value; this creates a high degree of persistence. Table 9 shows the results. In Fig. 10, we see that all of the correlations in the HP filtered data lie within the model’s 95 % confidence interval. Admittedly, this is not much of an accomplishment after several lags, since the confidence interval is very wide here. However, the confidence intervals are much tighter when there are only a few lags, so it is not entirely easy for the model to pass this test. The graph in Fig. 11 is similar to the one in Fig. 10, except that now I use the Hamilton filter instead of the HP filter. Fig. 12 depicts the forecast error variance decomposition for GDP in the model. Recall that there are three shocks: government spending, consumption sector labor productivity shocks, and the sunspot. The results are not sensitive to the number of lags; I chose six. Government spending shocks hardly explain any of the fluctuations in GDP. This is not a surprise; as in most RBCs, government

Table 6 Standard deviations and correlations for the model and the data, 1983–2007. Standard errors of the simulation results are in parentheses. Model

Y I C M2

Data z

0.0158 (0.0028) 0.0658 (0.0135) 0.0140 (0.0034) 0.0099 (0.0011)

13

HP filtered

Hamilton filtered

0.0106

0.0257

0.0492

0.1024

0.0083

0.0223

0.0099

0.0359

Journal of Economics and Business 109 (2020) 105891

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Table 7 Correlation coefficients in the model and the data for 1983–2007. The data is filtered GDP and M2. Standard errors are in parentheses. Correlation with M2t

Yt Yt+1 Yt+2 Yt+3 Yt+4 Yt+5 Yt+6 Yt+7 Yt+8

Model

HP Data

Hamilton Data

−0.1944 (0.1123) −0.2038 (0.0921) −0.1773 (0.0641) −0.1213 (0.0450) −0.0478 (0.0789) 0.0249 (0.0813) 0.0879 (0.0941) 0.1304 (0.0942) 0.1478 (0.0772)

−0.3274

−0.3788

−0.1669

−0.2656

−0.0363

−0.1885

0.0475

−0.1062

0.0937

−0.0492

0.1181

−0.0348

0.1548

0.0148

0.1673

0.0073

0.1937

−0.0403

Table 8 Correlation coefficients in the model and the data for 1983–2007. The data is filtered GDP and M2. Standard errors are in parentheses. Correlation with M2t

Yt Yt-1 Yt-2 Yt-3 Yt-4 Yt-5 Yt-6 Yt-7 Yt-8

Model

HP Data

Hamilton Data

−0.1944 (0.1123) −0.2767 (0.0921) 0.0896 (0.0641) 0.2433 (0.0450) 0.1923 (0.0789) 0.3594 (0.0813) 0.3612 (0.0941) 0.3222 (0.0942) 0.2656 (0.0772)

−0.3274

−0.3788

−0.3420

−0.3919

−0.2646

−0.3397

−0.1429

−0.2398

−0.0357

−0.1498

0.0814

−0.0729

0.1936

0.0341

0.2764

0.1190

0.3377

0.1769

spending does not have much of an impact. Back in Fig. 2, we saw that GDP increased by less than 0.25 % in response to a 1 % government spending shock. Unanticipated money shocks – i.e., the sunspot – explain more than half of the fluctuations in GDP. This is noticeably larger than most empirical estimates. However, that is because there are just three shocks in the model. If there were more sources of fluctuations, then each individual shock would play a smaller role. In Smets and Wouters (2005), monetary and productivity shocks are about equally important in explaining the variance in GDP. Thus, the importance of monetary shocks relative to productivity shocks in the model is in line with empirical estimates. Future research may explore a model with a larger number of shocks. To illustrate the idea of money as a sunspot variable as cleanly as possible, this paper relies on a more parsimonious approach.

8. Conclusion It has long been known that indeterminate models can have sunspot solutions, and it is easy to understand intuitively that selffulfilling prophecies can exist. However, it is hard to claim that sunspot variables are a feature of the real world if we are unable to identify a specific sunspot. The model addresses this criticism by using money as a sunspot variable. The model’s results are surprisingly consistent with the data. Thus, an RBC model can replicate several features of the data without sacrificing its key 14

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Fig. 8. Correlations between M2 and Y. The points are the correlations in the HP data; the shaded region is the 95 % confidence interval in the model.

Fig. 9. Correlations between M2 and Y. The points are the correlations in the Hamilton filtered data; the shaded region is the 95 % confidence interval in the model. Table 9 Autocorrelation in the model and the data for 1983–2007. The data is HP filtered log GDP. Standard errors are in parentheses. Correlation with M2t

Yt Yt-1 Yt-2 Yt-3 Yt-4 Yt-5 Yt-6 Yt-7 Yt-8

Model

HP Data

Hamilton Data

1.0000 (0.0000) 0.8543 (0.0405) 0.6598 (0.0868) 0.4485 (0.1390) 0.2510 (0.1912) 0.0915 (0.2352) −0.0125 (0.2652) −0.0570 (0.2756) −0.0465 (0.2674)

1.0000

1.0000

0.9025

0.9095

0.7671

0.7898

0.5760

0.6602

0.4065

0.5001

0.1855

0.3166

−0.0456

0.1527

−0.1962

−0.0070

−0.3365

−0.1600

15

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Fig. 10. The points are the correlations in the HP data; the shaded region is the 95 % confidence interval in the model.

Fig. 11. The points are the correlations in the Hamilton filtered data; the shaded region is the 95 % confidence interval in the model.

Fig. 12. Forecast error variance decomposition for GDP in the model’s simulations.

assumptions of flexible prices and no nominal rigidities. Overall, the model is compatible with many parts of the empirical data and introduces new techniques for using sunspots and explaining the effects of money. The model captures a number of stylized facts in the data in spite of several limitations. Even though there is no direct role for anticipated money and no unemployment, the model nonetheless matched the correlations in 1983–2007 quite well. The standard deviations and correlation coefficients in the model do not deviate too far from their empirical counterparts. Frictions and rigidities are absent from the model for a reason. I was not seeking to construct a model of money that is as realistic as possible. Rather, the intention was to explore whether the key features of the RBC model are truly irreconcilable with the monetary data. There are many extensions of the model that future research can study. One path forward is to incorporate frictions as well as money sunspots. This would allow anticipated money to have an impact. The money sunspot may amplify or diminish the effects of monetary policy, possibly allowing the model to explain the data better. This paper does not attempt to add frictions. In such a model, it may be unclear which factor (sunspot or frictions) is driving the results. The sunspot model without frictions might not be a special case of the more general model. There may be calibrations in which a sunspot solution exists when the frictions are present but not when they are absent. Kiley (1997) points out that this issue comes up frequently. The frictionless model in this paper presents the 16

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idea in its purest form: can we explain the monetary data solely with a sunspot variable, unassisted by other factors? The initial results are very promising and provide a stepping stone to further research. Acknowledgements This project received funding from the Kleinsorge Research Fellowship. George Evans, Bruce McGough, two anonymous referees, and seminar participants at the University of Oregon Macro Group provided helpful feedback and discussion. Any remaining errors are my own. Appendix A The Solow Residual From the production function for consumption goods, we know

Ct + Gt = Act(1 − α )(1 + θ) ((ut K ct )α Lct1 − α )1 + θ

(66)

However, K ct , Lct , and ut are not available in the data, so use the equations in the model to substitute them out: 1/ γ

αY ut = ⎜⎛ t ⎟⎞ ⎝ pt Kt ⎠

K ct = Kt ⎛ ⎝ ⎜

(67)

Ct + Gt ⎞ Yt ⎠ ⎟

(68) 1

Lct = (Ct + Gt ) ⎛ ⎝

⎜

(1 − α ) Ytσ ⎞ 1 − σ Nt zCt ⎠ ⎟

(69)

Simplifying:

Act(α − 1)(1 + θ)

⎡

= (Ct + Gt

α )θ ⎢ ⎛ ⎜

α/γ

⎞ ⎟

⎢ ⎝ pt ⎠ ⎣

−α + α / γ

⎛ Yt ⎞ ⎝ Kt ⎠

⎜

⎟

1

1−α 1+θ

⎤ ⎥ ⎥ ⎦

⎛ ⎛ (1 − α ) Ytσ ⎞ 1 − σ ⎞ Nt ⎟ ⎜ zCt ⎠ ⎠ ⎝⎝ ⎜

⎟

(70)

Recall that pt is not the CPI; it is the price of investment goods relative to consumption goods. This is also not available in the data, so we also substitute it out. Taking the ratio of the production functions, we find that θ

C + Gt ⎞ ⎛ Act ⎞ pt = ⎜⎛ t ⎟ ⎝ pt It ⎠ ⎝ AIt ⎠ ⎜

(1 − α )(1 + θ )

⎟

(71)

Note that pt It does exist in the data, since it corresponds to the investment component of GDP.

Act(α − 1)(1 + θ)

⎡

= (Ct + Gt

C )θ ⎢α α / γ ⎛ t ⎜

⎢ ⎣

⎝

−αθ

+ Gt ⎞ ⎟ pt It ⎠

−α (1 − α )(1 + θ )

⎛ Act ⎞ ⎝ AIt ⎠

⎜

⎟

−α + α / γ

⎛ Yt ⎞ ⎜ ⎟ ⎝ pt Kt ⎠

1

1−α 1+θ

⎛ ⎛ (1 − α ) Ytσ ⎞ 1 − σ ⎞ Nt ⎟ ⎜ zCt ⎠ ⎠ ⎝⎝ ⎜

⎟

⎤ ⎥ ⎥ ⎦

(72)

Quarterly capital data does not seem to be available, but annual data is. Fortunately, the capital to output ratio is very stable over time, so my assumption that it is constant within each year should have a negligible impact. Note that Act and AIt cannot be separately identified except in special cases; there are two unknowns and only one equation. “But can’t you just use the production function for the investment sector in the same way you did for the consumption sector? That would give you a second equation,” you might ask. The answer is no. That is because the production function for the investment sector was already used in order to substitute out pt . I can see three special cases in which the shocks can be identified, even though there is just one equation for two unknowns. (1) ∼ ∼ ∼ ∼ Act = AIt , (2) Act = 0 , and (3) AIt = 0 . It is arguably unrealistic to shut down the shocks in one of the sectors, but the only alternative seems to be assuming that the shocks are always identical in both sectors – also unrealistic. ∼ I picked the third case: AIt = 0 for all t. Here’s why. First of all, the model is far too volatile in the other scenarios. There is ∼ ∼ intuition for this. We will start with the first case, Act = AIt . Recall why it is easier to obtain indeterminacy in two-sector models or models with variable capital utilization. These features amplify the increasing returns, thus lowering the degree of increasing returns necessary to achieve indeterminacy. However, these aspects of the model also amplify the economy’s response to labor productivity shocks. Therefore it is not surprising that the IRFs were highly volatile when I experimented with shocks in both sectors. A very sensitive IRF may be tolerable if the variance of shocks is very low. However, for simulations in the model with shocks in both sectors, the variance of the Solow residual was never low enough to yield realistic estimates of the standard deviation of output. If Case 1 doesn’t work, then the shocks have to be shut down in one of the sectors. To dampen the volatility, it is intuitively clear that having shocks only in the consumption sector is better than having shocks only in the investment sector. This is because 17

Journal of Economics and Business 109 (2020) 105891

M.S. Wilson

Table A1 Regression results. Independent variables

F stat

P-value

8 lags of the sunspot 8 lags of labor 8 lags of labor, CPI, GDP, M2, gov shocks, tech shocks 4 lags of the sunspot 5 lags of the sunspot 6 lags of the sunspot 7 lags of the sunspot Sunspott-4 Expected change in Fed funds rate (Romer & Romer, 2004) Expected change in Fed funds rate (Kuttner, 2001)

0.88 0.60 0.54 1.53 1.63 1.61 1.01 3.54 2.43 4.19

0.5356 0.7716 0.9845 0.2005 0.1597 0.1540 0.4273 0.0630 0.1247 0.0522

investment sector goods become capital which is used to produce consumption goods, while consumption goods are not used at all to make investment goods. Hence the model will respond more to investment sector shocks than to consumption sector shocks. Empirical papers have attempted to identify productivity shocks, but there a problem with trying to use those estimates in my project. They will be inconsistent with the production function in the model. That is why I have stuck to the Solow residual. Setting ∼ AIt = 0 for all t is the best approach. ∼ Now we can solve for Act . 1

1−α

−α + α / γ −αθ ⎛ ⎛ ⎛ (1 − α ) Ytσ ⎞ 1 − σ ⎞ 1 C + Gt ⎞ ⎛ Yt ⎞ log ⎜ (Ct + Gt )θ /(1 + θ) α α / γ ⎜⎛ t Nt ⎟ ⎟ ⎜ ⎟ ⎜ α−1 zCt ⎜ ⎠ ⎝ pt It ⎠ ⎝ pt Kt ⎠ ⎠ ⎝⎝ ⎝ ∼ 1 . We can then run the regression Thus, Act is the residual multiplied by 1 − α (1 + θ ) ∼ ∼ Act = ρac Ac, t − 1 + εac, t ⎜

⎟

⎞ ∼ ⎟ = A¯ + gA t + (1 − α (1 + θ)) Act ⎟ ⎠

(73)

(74)

and identify the variance of labor productivity shocks. I calibrate ρac = 0.95 based on the regression results. Appendix B Testing the conditional mean zero assumption To test the assumption that the sunspot has a conditional mean of zero, I regressed the sunspot on lags of the variables in the model, including lags of the sunspot. I never rejected the null hypothesis that all the variables were insignificant. One variable (sunspott-4) was barely significant at the 5 % level when 8 lags of the sunspot were used. To investigate this further, I ran additional regressions with varying lags of the sunspot. The null hypothesis was never rejected in these regressions. These results are consistent with the assumption that the sunspot has a conditional mean of zero. Table A1 presents the outcomes of these regressions. Appendix C Proofs PROPOSITION.

α (Yct + pt YIt ) ut

= pt utγ − 1 Kt

PROOF. Define rt = + rtu ut . For each unit of capital, firms pay rtk and then they also pay rtu for each unit of ut . Thus, the household budget constraint is Ct + pt It + Gt = (rtk + rtu ut ) Kt + wt Lt . The first order conditions from the Bellman system are as follows.

rtk

∂V (wt , rtk , pt , kt , gt , rtu ) ∂lt ∂V (wt , rtk , pt , kt , gt , rtu ) ∂it ∂V

(wt , rtk ,

pt ,

∂ut

kt , gt , rtu )

=

wt − zlt−σ = 0 ct

(75)

=

−pt + βe nEt Vk (wt + 1, rtk+ 1, pt + 1 , kt + 1, gt + 1, rtu+ 1 ) e−n = 0 ct

(76)

=

rtu kt − βe nEt Vk (wt + 1, rtk+ 1, pt + 1 , kt + 1, gt + 1, rtu+ 1 )(e−nkt utγ − 1) = 0 ct

(77)

The envelope condition is below.

Vk (wt , rtk , pt , kt , gt , rtu ) =

p (1 − δt ) rtk + rtu ut + t ct ct

(78)

18

Journal of Economics and Business 109 (2020) 105891

M.S. Wilson

Thus, the households supply curve for ut is

rtu = pt utγ − 1

(79)

The firms’ first order conditions for ut are

αYct − rtu K ct = 0 ut

(80)

αpt YIt − rtu KIt = 0 ut

(81)

When combined, aggregate demand for ut is as follows.

α (Yct + pt YIt ) = rtu Kt ut Substitute in for

(82)

rtu :

α (Yct + pt YIt ) = pt utγ − 1 Kt ut

(83)

This proves the proposition.

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