Macroeconomic Dynamics, 18, 2014, 338–368. Printed in the United States of America.
doi:10.1017/S1365100512000417

CREDIT CONSTRAINTS, LEARNING,
AND AGGREGATE CONSUMPTION
VOLATILITY
DANIEL L. TORTORICE
Brandeis University

This paper documents three facts. First, the volatility of consumption growth relative to
income growth rose from 1947 to 1960 and then fell dramatically by 50% from the 1960s
to the 1990s. Second, the correlation between consumption growth and personal income
growth fell by about 50% over the same time period. Finally, the absolute deviation of
consumption growth from its mean exhibits one break in U.S. data, and the mean of the
absolute deviations has fallen by about 30%. A standard dynamic stochastic general
equilibrium model is unable to explain these facts. I examine ability of two hypotheses to
account for these facts: a fall in credit constraints and changing beliefs about the
permanence of income shocks. I find evidence for both explanations. The beliefs
explanation is more consistent with the data.
Keywords: Consumption, Business Fluctuations, Learning, Liquidity Constraints

1. INTRODUCTION
This paper establishes three facts about aggregate U.S. consumption. First, after
rising from the 1950s to the 1960s, the standard deviation of consumption growth
relative to income growth fell 40% from the 1960s to the 1970s and then another
30% from the 1980s to the 1990s. Finally, it rose about 25% during the most
recent recession. Second, the correlation between consumption growth and income
growth has fallen 50% over this same period. Finally, there is a structural break in
the size of the absolute deviation of consumption growth from its mean. The break
is in 1977, and the total fall in the absolute deviation of consumption growth from
its mean is 30%. These facts complement the Great Moderation facts by showing
that consumption volatility has fallen more than might have been expected based
on the fall in income volatility alone.

A previous version of this paper was titled “Income Process Uncertainty, Learning and Aggregate Consumption
Volatility.” I thank Robert Barro, John Campbell, Gita Gopinath, George Hall, David Laibson, N. Gregory Mankiw,
Catherine Mann, Blake LeBaron, James Stock, Aleh Tsyvinski, two anonymous referees, and attendees of the Harvard
macroeconomics lunch seminar, the Brandeis faculty workshop, and the Society for Economic Dynamics 2010 Annual
Meeting for helpful comments, suggestions, and advice. I am grateful to the Jacob K. Javits Fellowship Program at
the U.S. Department of Education, Harvard University, and the Brandeis University Theodore and Jane Norman Fund
for financial support. All errors are mine. Address correspondence to: Daniel L. Tortorice, Department of Economics,
Brandeis University, Mailstop 021, 415 South St., Waltham, MA 02454-9110, USA; e-mail: tortoric@brandeis.edu.
c 2012 Cambridge University Press


1365-1005/12

338

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

339

This paper joins a growing literature suggesting a changing relationship between consumption and income. Stock and Watson (2003) estimate structural
breaks in consumption. Dynan et al. (2006, 2009) note that the marginal propensity to consume has fallen substantially. This paper, however, is the first to
document all three outlined facts and study two different hypothesis: a reduction in credit constraints and changing beliefs about the productivity process as
explanations.
I model credit constraints as in Ludvigson (1999), in which impatient consumers
face a binding debt limit. I estimate a fraction of credit-constrained consumers that
varies over time and use this model to simulate a consumption series. I examine
how well this series replicates the stated consumption facts.
It is not obvious that reduced credit constraints reduce the volatility of consumption. If individuals want to smooth income shocks, then reducing financial constraints makes consumption smoother. However, if individuals want to
borrow to respond more than one-to-one with income shocks—for example, if
shocks are permanent—then reduced financial constraints could increase consumption volatility. I therefore model uncertainty as to the permanence of income
shocks.
As pointed out by Deaton (1992), if income shocks are persistent but transitory,
consumption is less volatile than income. However, if shocks are permanent, consumption may be more volatile than income. Cochrane (1988) and Stock (1991)
show that distinguishing between these two models in samples the length of the
U.S. macroeconomic time series is very difficult. Because these two income processes are difficult to distinguish, one might expect consumers to have uncertainty
as to which is the true model. Their beliefs may change over time, resulting in
time-varying consumption-to-income volatility.1
To examine the plausibility of this hypothesis, I study a learning model based
on Cogley and Sargent (2005). My model is a standard dynamic stochastic general equilibrium model in which output fluctuations are driven by changes in
productivity, with one modification. The agent believes the productivity process
is nonstationary with some probability and stationary otherwise. At each point
in time the individual first updates her beliefs about the parameters of the two
models, and then, using Bayes’s rule, updates her beliefs about the probability
that each model is true. Based on these beliefs, she chooses her optimal level of
consumption. From this model I can simulate a consumption series and examine
the model’s ability to capture the previously outlined facts.
I find that two benchmark dynamic stochastic general equilibrium (DSGE)
models with a known productivity process fail to predict the fall in consumption
volatility. The learning model, however, is able to partly explain the rise and fall in
consumption volatility early in the sample and also the fall and rise in consumption
volatility toward the end of the sample. The credit constraints model is consistent
with the early rise and fall in consumption volatility, but cannot explain the
magnitude of the fall in consumption volatility. Importantly, only a model with
learning about the productivity process can replicate the estimated break. Finally,

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

340

DANIEL L. TORTORICE

the learning model’s implied probability weights on the different productivity
processes have significant explanatory power for consumption changes.
The learning model also has an implication for recent economic events. During
the current recession, the weight on the nonstationary model has increased substantially; the model implies a large drop in consumption, consistent with the data.
This is perhaps one of the most interesting predictions of the learning model.
This paper relates to three strands of literature. First, it relates to the literature on
learning in macroeconomic models [Evans and Honkapohja (2001); Georges and
Wallace (2009)]. It shows how relaxing the strict rational expectations assumption
in favor of learning results in a significantly different optimal policy. In my learning
model, the two possible productivity processes are difficult to distinguish in small
samples. Hence, learning which process is true takes a long time. However, because
consumption volatility is sensitive to deviations from stationarity, the processes
imply different optimal policies. Learning then introduces a very different choice
of consumption than in the no-learning model. As in Waters (2009) and Tillman
(2011), the inclusion of uncertainty changes the optimal policy.
The paper also relates to some recent research on consumption. Guvenen (2007)
finds that introducing learning into a life-cycle consumption model changes many
of the model’s predictions. Specifically, optimal consumption choices are very
different when one learns about the trend in income than when it is known. The
paper also relates to Aguiar and Gopinath (2007). They explain the differences
in consumption volatility relative to income volatility across countries. In their
model, different countries have different ratios of consumption volatility to income
volatility because they have different productivity processes. In my paper, the
variance of consumption relative to income varies over time depending on the
relative likelihood of two different productivity processes.
Finally, this paper relates to the Great Moderation literature. Although an extensive review would be out of place, I note three closely related papers. Cecchetti
et al. (2005) show that the reduction in consumption volatility was accompanied
by a rise in U.S. debt levels, supporting a financial innovation explanation. Dynan
et al. (2006) show that marginal propensities to consume have fallen over time.
Cecchetti et al. (2006) relates changes in consumption volatility across countries
to changes in estimated fractions of rule-of-thumb consumers.
This paper differs for three reasons. First, the question is different. I ask if financial innovation can account for the fact that consumption volatility has declined
more than personal income volatility in U.S. data. Second, I simulate a model
of credit constraints and test the model directly to see if it is consistent with the
decline in consumption volatility. Finally, I solve an innovative learning model,
and it explains features of the data that the credit constraints model does not.
The rest of the paper proceeds as follows. Section 2 describes the data I use.
Section 3 describes the empirical facts I attempt to explain. Section 4 describes the
credit constraint and learning models. Section 5 tests these models, and Section 6
provides more intuition for the learning model’s predictions. Section 7 examines
robustness to different parameter choices. Section 8 concludes.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

341

2. DATA
Data come from the National Income and Product Accounts (NIPAs). The consumption data are aggregate consumption of nondurable goods and services, and
income is the personal income series. Data begin in 1947 and end in 2010:1.2
Series are transformed into real per capita series by deflating with the nondurable
consumption deflator, the service consumption deflator, and the GDP deflator,
respectively, and dividing by population.
I pay particular attention to the distinction between accrued wages and distributed wages. In general, firms distribute most of the wages accrued or earned
within a quarter. However, at times firms distribute wages early or late. Occasionally this difference is large. For example, in 1992:Q4 firms distributed 63
billion dollars more in wages than employees earned, and in 1993:Q1 employees
earned 72.1 billion dollars more in wages than firms distributed. These discrepancies appear because of increases in income tax rates.3 These distributions are
large enough to affect learning about the productivity process. They appear as
large shocks to income, though they do not represent uncertainty. As a result,
I assign to income all wages earned in a quarter regardless of how much is
distributed. However, this modification does not affect the conclusions of the
paper.

3. EMPIRICAL FACTS
This section establishes three facts about the aggregate U.S. consumption series.
First, the volatility of consumption relative to income rose early in the sample and
then fell substantially over the last 45 years. Second, the correlation of consumption
growth and income growth has fallen over time. Finally, there is a substantial break
in the absolute deviation of consumption growth from its mean, falling about 30%
over the last 60 years.

3.1. Consumption Volatility Relative to Income Volatility
In Figure 1, I plot the rolling standard deviation of consumption growth divided by
the rolling standard deviation for income growth. I use a window over the next 10
years and nondurables and services consumption.4 Results are robust to varying
the window from 20 to 50 quarters. As one can clearly see, consumption variance
relative to income variance rises over time, reaching a peak around 1960, and then
falls quite dramatically by 40% from 1960 to 1970. It remains constant for about
20 years before beginning to fall again in 1990 by an additional 30%. During the
most recent recession it increased by 25%. I obtained similar results (omitted)
with nondurables alone, finding a similar, but more gradual decline beginning in
1960 and ending in 1980.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

342

DANIEL L. TORTORICE
1.2

1

0.8

0.6

0.4

0.2

1950

1960

1970

1980

1990

2000

FIGURE 1. Ratio of the standard deviation of consumption growth to income growth. Dashed
lines represent the 95% confidence interval.

3.2. Time-Varying Response of Consumption to Income
The data demonstrate that consumption changes appear to have moderated over
time relative to income changes. To get a parametric representation of this fact, I
estimate a time-varying coefficient in the regression
 ln ct = α + mt  ln yt + εt ,

(1)

where ct is the NIPA value of real per capita nondurable and services consumption
and yt is real per capita personal income. I let mt take the form mt = m0 + m1 t.5
I found that higher-order terms (quadratic and cubic) were not significant. Consumption and income growth are measured as annualized percentage changes.
The results, in column (1) of Table 1, indicate that there is a significant timevarying component to m, as m1 is statistically significant and negative. At the
beginning of the sample period mt = 0.55; by the end of the sample period
mt is around 0.276. This result again shows a considerable moderation of consumption relative to personal income. Although income has moderated over time,
consistent with the Great Moderation facts, consumption has moderated even
more.
This approach is one way of measuring a marginal propensity to consume.
Dynan et al. (2009) estimate a marginal propensity to consume, controlling for
potential output, interest rates, the unemployment rate, and the index of consumer
sentiment. They also find a substantial fall in the response of consumption to
income.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

343

TABLE 1. Relationship between income growth and consumption growth
Benchmark models
Data
0.55∗∗∗
[0.08]
−0.124∗∗∗
 ln(inc.)∗
(time/100) [0.043]
Constant
1.17∗∗∗
[0.19]
Obs.
221
 ln(inc.)

Nonstationary Stationary Learning
1.03∗∗∗
[0.04]
0.021
[0.021]
−0.11
[0.08]
221

0.50∗∗∗
[0.03]
−0.009
[0.017]
1.17∗∗∗
[0.05]
221

Credit

Credit +
learning

0.70∗∗∗
0.73∗∗∗
0.81∗∗∗
[0.04]
[0.02]
[0.02]
−0.073∗∗∗ −0.050∗∗ −0.061∗∗∗
[0.025]
[0.024]
[0.012]
0.90∗∗∗
0.76∗∗∗
0.60∗∗∗
[0.08]
[0.07]
[0.04]
221
221
221

Note: This table reports the result of regressing annualized consumption growth from the data and the predictions
of the different models on the annualized income growth rate, allowing the coefficient to vary by time.
Newey–West standard errors with five lags in brackets. ∗ Significant at 10%; ∗∗ Significant at 5%; ∗∗∗ Significant at
1%.

3.3. Breaks in the Variance of Consumption
Because consumption has moderated more than income volatility, it is natural
to examine how much consumption has moderated. To investigate, I follow the
methodology of Stock and Watson (2003) and estimate a break in the mean of
the absolute value of the residual from the regression of nondurable consumption
growth on a constant.6
 ln ct = α + ηt ,
|ηt | = ζ + ζ1 τ + εt .

(2)
(3)

To estimate τ (the break dates), I use the methodology of Bai and Perron (1998)
and the algorithms and GAUSS code available in Bai and Perron (2003). The
methodology, in addition to providing a feasible way to estimate the structural
break model, describes a sequential method for estimating the number of breaks
(allowing for the possibility of more than one break). These methods consistently
estimate the number of breaks and the proportion of the sample that occurs before
the break date.
To estimate the number of breaks, Bai and Perron (1998) recommend the sequential application of a test of l breaks versus the alternative of l +1 breaks, called
the sup F (l + 1|l) test. The procedure first tests one break versus the alternative
of zero breaks, and then two versus one, and so on until the statistic fails to reject
the null hypothesis. In addition, Bai and Perron provide a test statistic, called the
U D max statistic, that tests the hypothesis of zero breaks against the alternative
of k > 0 breaks, where k is unknown. The estimates of the break dates are those
that minimize the sum of squared residuals in equation (3).

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

344

DANIEL L. TORTORICE

TABLE 2. Breaks in consumption volatility
Benchmark models
Data

Nonstationary Stationary

Learning

Credit

Credit +
learning
7.29∗
3.86

supF(1|0)
supF(2|1)

7.38∗
3.44

2.32
5.24

3.2
4.64

9.47∗
3.71

5.26
4.33

UD Max

7.38

6.07

4.4

10.61∗

5.26

7.29

Break
90% CI

1977:4
1957:2–1993:1

—
—

—
—

1982:1
1973:2–1995:3

—
—

1984:3
1971:2–2000:2

2.65
(0.20)
1.87
(0.17)

—
—
—
—

—
—
—
—

1.5
(0.11)
0.94
(0.11)

—
—
—
—

1.77
(0.12)
1.21
(0.13)

δ1
δ2

Note: This table reports the results from testing for a break in the residual of the annualized consumption change
regressed on a constant. The means of the residual are given by δ, for before and after the breaks. supF and UDMax
are tests for L+1 breaks vs. L and any breaks respectively and the critical values for the tests from Bai and Perron
(2003) are given as follows. (star denotes >= 10% significance):
Critical values for tests
10%
supF(1|0)
supF(2|1)
supF(3|2)
UD Max

5%

1%

7.04 8.58 12.29
8.51 10.13 13.89
9.41 11.14 12.66
7.46 8.88 12.37

Table 2 presents the results from these tests.7 Consumption growth is measured
as an annualized percentage change. The consumption series has one break: at
1977:4. The means for each segment are 2.65 and 1.87. This implies a moderation
in consumption volatility of 30%.
4. EXPLANATIONS
This section outlines the two models I use to explain the fall in consumption
volatility relative to income volatility. The first model is a model of credit constraints with a time-varying fraction of credit-constrained consumers. The second
is a model of learning in which the true productivity process is unknown and
therefore agents’ beliefs about the true process are changing over time.
4.1. Credit Constraints
This section of the paper outlines a model with time-varying credit constraints as an
attempt to explain the previously described empirical facts. For another example
analyzing the interaction of credit constraints and consumption see Benito and
Mumtaz (2009). Consumption may be very volatile early in the sample because

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

345

individuals are credit-constrained, but become less volatile as more assets are
accumulated or better consumption insurance becomes available. To examine this
explanation, I assume that aggregate consumption comes from two types of agents:
an unconstrained agent and a constrained agent:
ct = λt ctcc + (1 − λt )ctuc .

(4)

Here λt is the fraction of the population that is credit-constrained, ctcc is the
change in consumption of the credit-constrained consumers, and ctuc is the consumption change of the unconstrained consumer. I model the constrained consumer
as in Ludvigson (1999). In this model the agent chooses consumption to maximize
 1−γ 
∞

Ct+i
Et
βi
(5)
1
−γ
i=0
subject to Dt+1 = (1 + r)(Dt + Ct − Yt ), where Dt is consumer debt, r is
the real interest rate, and Yt is labor income. Labor income evolves according to
 ln Yt = g +ηt −ψηt−1 , where ηt is i.i.d. Consumers are credit-constrained: they
can borrow only up to D t+1 = w1 Yt exp(ξt ), where ξt+1 = φξt + ut+1. Therefore,
−Dt (1+r)
+ Yt .
maximum consumption at time t is given by Xt = Dt+1(1+r)
zt+1 = Yt+1 /Yt , we can solve for the policy
If we define wt = Xt /Yt and 
function θt (wt , ηt , ξt ) = Ct /Yt , as a function of the state variables, by iterating on the Euler equation. Defining the marginal-utility-of-consumption function
−γ
pt (wt , ηt , ξt ) = v(θt ) = θt , Ludvigson (1999) shows that


−γ
zt+1 p(wt+1 , ηt+1 , ξt+1 ) ,
(6)
pt (wt , ηt , ξt ) = max v(wt ), β ∗ Et

−1
t+1 −ψηt +ξt+1 −ξt +g)−(1+r)
zt+1
(1+r)[wt −v −1 pt (wt , ηt , ξt )]+ exp(η
where wt+1 = 1+
(1+r) exp(ηt+1 −ψηt +g+ln ω−ξt )
∗
and β = (1 + r)β. I solve for pt (wt , ηt , ξt ) by iterating on (6) using five grid
points for η and ξ . I discretize the i.i.d. ηt and the AR(1) ξt using the method
of Tauchen and Hussey (1991).8 Finally, I use 100 grid points for w and linear
interpolation to calculate the function between grid points of w.
To calculate the predicted aggregate consumption change I need to estimate λt .
To do so let ctuc = μ + σ rt + εt , where rt is the ex post real interest rate and
εt is i.i.d., because the unconstrained consumer does not respond to predictable
p
variation in income, and let ctcc = α + σ c rt + mpct yt + νt where vt is
p
i.i.d., yt is the predicted change in income, and mpct is the credit-constrained
individual’s marginal propensity to consume out of predictable income estimated
from the model.9 Then
p

or

ct = λt (α + σ c rt + mpct yt + νt ) + (1 − λt )(μ + σ rt + εt )

(7)


	



p
ct = μ + σ rt + λt mpct yt + σ c − σ rt + α − μ + ηt .

(8)

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

346

DANIEL L. TORTORICE
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1950

1960

1970

1980

1990

2000

2010

FIGURE 2. Estimated fraction of credit-constrained consumers.

p

To estimate this equation, I first calculate yt as the fitted value of a regression
of yt on the lagged income growth rate, the lagged change in the S&P 500 index,
and the lag ex post real interest rate; then I multiply by the estimated mpct .10
Because the ex post real interest rate is also correlated with ηt , I instrument its
value with the same instruments used for income growth to obtain 
rt . The final
equation I estimate is

	



p
rt + λt mpct yt + σ c − σ 
rt + α − μ + ηt .
ct = μ + σ

(9)

I estimate the time-varying λt using the Kalman filter [see Hamilton (1994,
pp. 399–402)].11 Figure 2 plots the resulting series for the estimated fraction of
consumers who are credit-constrained. Credit constraints rise in the 1960s, with a
peak of 80% credit-constrained. Credit constraints fall nonmonotonically to a low
of 10% in 2010, though most of that fall is in the last few years. As late as 2005,
40% of the population was still estimated to be credit-constrained.
I follow Campbell and Mankiw (1990) and estimate the fraction of creditconstrained consumers using data from 1955 onward. This avoids the Korean War
period and some large income swings due to government transfer payments. To
estimate credit constraints pre-1955, I assume that credit constraints do not change
from 1947 to 1955. This assumption is only for completeness and does not affect
the results.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

347

4.2. The Learning Model
This section describes a model economy in which the representative agent maximizes consumption while also investing in a capital stock. The agent combines
capital and labor to produce output. Productivity grows at a constant rate g, but
is subject to stochastic shocks. The agent is learning if the correct model for the
productivity shocks is one with permanent or temporary deviations from trend,
and also about the parameters of each productivity model.
Model description.
mizes

For the learning model, the representative agent maxi-

max

{ct+j ,lt+j ,it+j }∞
j =0

Et

∞

j =0

βj

{ct+j [1 − (θ lt+j )φ ]}1−γ
1−γ

(10)

subject to, with probability p = ps,t ,
kt+1 = (1 − δ)kt + it ,
ktα [A0 (1

+ g) e lt ]
t zt

1−α

= ct + it ,

zt = θ1S zt−1 + ... + θpS zt−p + εts ,

(11)
(12)
(13)

and with probability p = pns,t = 1 − ps,t ,
kt+1 = (1 − δ)kt + it ,
ktα [A0 (1

+ g) e lt ]
t zt

1−α

= ct + it ,

zt = θ1NS zt−1 + ... + θpNS zt−p+1 + εtns .

(14)
(15)
(16)

Here kt is beginning-of-the-period capital stock, yt is output, ct is consumption,
it is investment, lt is labor supply, and A0 (1 + g)t ezt is the productivity level. With
probability ps,t , the log deviation of productivity from its trend is given by (13)
and is therefore stationary around the time trend; with probability pns,t , the log
deviation of productivity from its trend is given by the unit root process (16) and
hence exhibits permanent deviations from the trend.12,13
Updating of beliefs. To choose consumption, the agent each period must
update his beliefs concerning the probability that each model is true. I update
beliefs as in Cogley and Sargent (2005).
There are two models of the productivity process, indexed i = s, ns, which can

be written in regression form as yi,t = xi,t θi,t + εi,t . For the stationary model the
regression equation is (13); for the nonstationary model the equation is (16).
t represent all observations on yt and xt up to time t, the agent’s prior
Letting Z
beliefs on the parameters for each model are given by


	
−1
t−1 ) = N θt−1 , σ 2 Pt−1
,
(17)
p(θ |σ 2 , Z
p(σ 2 |Z t−1 ) = IG(st−1 , vt−1 ).

(18)

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

348

DANIEL L. TORTORICE

−1
θt−1 is the parameter estimate based on t − 1 data, σ 2 Pt−1
is the estimate of the
variance–covariance matrix of θt−1 , st−1 is the residual sum of squares, and vt−1
is the degrees of freedom for estimating variance of the residuals (t − k). IG is
the inverse gamma distribution. I follow Cogley and Sargent (2005) and use the
normal and inverse gamma distributions because these allow one to update the
parameters recursively and use analytical formulas to update the posterior beliefs.
Stepping out of the normal–inverse gamma family would involve computationally
intensive simulations at each point in time to calculate the likelihood (23) of each
model.
Bayesian updating implies that the parameters are updated recursively according
to

Pt = Pt−1 + xt xt ,

(19)

θt = Pt−1 (Pt−1 θt−1 + xt yt ),




(20)


st = st−1 + yt yt + θt−1 Pt−1 θt−1 − θt Pt θt ,

(21)

vt = vt−1 + 1.

(22)

Next the agent updates the probability weights on each model. For each model,
the marginalized likelihood is

 
 
t

 	


	
p ys |xs , θi , σi2 p θi , σi2 dθi dσi2
mit =

(23)

s=1

and the probability weight is wit = mit pi,0 , where pi,0 is the prior probability on
model i.
To calculate the marginalized likelihood, Cogley and Sargent (2005) note that
Bayes’s rule implies, for any θi , σi2 ,
t


	


t =
p θi , σi2 |Z

s=1

mit
t


mit =

p(ys |xs , θi , σi2 )p(θi , σi2 )

s=1

,

(24)

.

(25)

p(ys |xs , θi , σi2 )p(θi , σi2 )
t )
p(θi , σi2 |Z

Therefore,
wi,t+1
mi,t+1 pi,0
=
,
wi,t
mi,t pi,0
=

p(yt+1 |xt+1 , θi , σi2 )

(26)


	
t
p θi , σi2 |Z

.
	
t+1
p θi , σi2 |Z

(27)

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

349

As Cogley and Sargent (2005) show, this expression can be evaluated analytically,
because


	

p yt+1 |xt+1 , θi , σi2 = N (yt+1 − xt+1 θi , 0, σi ),


	
−1
, st+1 , vt+1 ),
p θi , σi2 |Zt+1 = NIG(θi , σi ; θt+1 , Pt+1

(28)
(29)

where N and NI G indicate the normal and the normal-inverse gamma probability
density functions, respectively. There are analytical expressions for both distributions, which are available in Cogley and Sargent (2005). Finally, to get the actual
probabilities for each model, one normalizes the weights to sum to one:
pNS,t =

wNS,t
,
wNS,t + wS,t

(30)

pS,t = 1 − pNS,t

(31)

Model solution. Now that we can calculate the agent’s beliefs at each point in
time, I can solve for the optimal consumption, investment, and labor supply plans.
I first normalize the trending variables by the nonstochastic level of productivity
and then take a linear–quadratic approximation to the utility function around the
nonstochastic steady state. Subsequently, I cast the model as a stochastic optimal
linear regulator problem and solve it using standard methods [e.g., of Ljungovist
and Sargent (2004, Ch. 5) and Cogley and Sargent (2005)].14
For a given variable st , letting 
st = st /(1 + g)t (normalizing A0 = 1) and
substituting the resource constraint into the utility function, we can rewrite the
objective as

max

{lt+j ,
it+j }∞
j =0

Et

∞

j =0

β

∗j

	

φ
α
{[
kt+j
(ezt lt+j )1−α − 
it+j ][1 − θ lt+j ]}1−γ
1−γ

,

(32)

where β ∗ = β(1 + g)1−γ , and rewrite the capital evolution equation as
kt + 
it .
(1 + g)
kt+1 = (1 − δ)

(33)

Next, I take a linear–quadratic approximation to the utility function. See the
Appendix for details.
Following Cogley and Sargent (2005), I write each submodel (i.e., the model for
each possible productivity process) as a linear regulator problem and then stack
the model matrices to cast the complete model as an optimal linear regulator.
For the stationary model, the state variables are xtS = [1 zt kt zt−1 ... zt−p+1 ]
and the control variables are ut = [lt it ] . For the nonstationary model, the state
variables are xtNS = [1 zt kt zt zt−1 ... zt−p+2 ] and the control variables are
ut = [lt it ] . Then for model i = {s, ns}, I can write the problem in the linear
Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

350

DANIEL L. TORTORICE

regulator format,
Uti = max∞ Et
{ut+j }j =0

∞


	 i i i


i
β ∗j xt+j
R xt+j + ut+j Qut+j + 2xt+j
W i ut+j ,

(34)

j =0

subject to
i
i
xt+1
= Ait xti + B i ut + C i εt+1
.
i

i

i

(35)

, Ait , B i , C i

The matrices R , Q , W
are given in the Appendix.
For the complete model, the agent seeks to maximize15
max ps,t Et Uts + (1 − ps,t )Et Utns .

(36)

{ut+j }∞
j =0

We can write this as an optimal control problem by stacking the matrices of the
submodels,
max∞ Et

{ut+j }j =0

∞





β ∗j (xt+j
Rxt+j
+ ut+j Qut+j + 2xt+j
W ut+j ),

(37)

j =0

subject to
xt+1 = At xt + But + Cεt+1 ,

(38)

where
xt =

[xts

xtns ] ,


0
ps,t R S
,
R=
0
pns,t R ns


 S
0
At
At =
,
0 ANS
t




ps,t W s
W =
,
pns,t W ns






Bs
B=
,
B NS




Cs
C=
.
C NS

5. TESTING THE MODELS EMPIRICALLY
5.1. Credit Model Simulation
To simulate a consumption series from the credit constraint model, I first need
to parameterize the model. I use the Ludvigson (1999) calibration adjusting for
1
quarterly data: ψ = 0.44, r = 0.03/4, β = 1+0.15/4
, φ = 0.6, ω = 15, γ =
u
y
2, σ = 0.0025, and σ = 0.025. I next need to simulate the time path for
household income. I first assume an economy with 5,000 households, all described
by the credit constraint model. As in Ludvigson (1999), the household income
process is decomposed into an aggregate shock and two idiosyncratic shocks, as in
h
h
h
+ ε3,t
− ε3,t
. Here, ε1 is the aggregate shock and
ηt − ψηt−1 = ε1,t + χ ε1,t−1 + ε2,t
h
h
ε2 and ε3 are household-specific shocks. We need two household shocks to match
observed positive autocorrelation in aggregate income and negative autocorrelation
in household income. I estimate χ and the ε1 time series from an ARIMA(0,0,1)
model on the real per capita personal income growth series. The ARIMA estimate
Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

351

also gives the growth rate of income g = 0.0052. I then estimate the variances of
the idiosyncratic shocks by matching the variance of the household income series
and its autocorrelation. I simulate the household shocks as random draws from
a normal distribution. Finally, to get a time series for ξt (the shock to the debt
limit), I draw a random chi from the grid according to the probabilities given by
the method of Tauchen and Hussey (1991).
Once I have each household’s consumption series, I can sum across households
to obtain an aggregate consumption series. I match this to the data by scaling up
the mean of this series to the mean of real per capita nondurables and services
consumption in the data. Then I combine consumption changes from this scaled
series with consumption changes from the learning model, using equation (4)
to generate the finalized consumption series from the credit constraint model. I
generate two series: a credit series that combines consumption from the credit
constraint model with consumption from the learning model when individuals
believe that the productivity process is stationary, and a credit and learning series
that combines consumption from the credit constraint model with consumption
from the learning model when individuals’ beliefs change over time.
5.2. Learning Model Simulation
The calibration of the learning model is fairly standard: β = 0.99, α = 13 ,
δ = 0.025, and γ = 2. A higher γ leads to more volatile consumption, and I
choose γ = 2 to better match the volatility of consumption. Finally, I choose
θ and φ so that the agent inelastically spends 1/3 of his time working. This
assumption, although not necessary to generate the observed learning dynamics,
allows me to choose productivity so that income in the model exactly matches
income in the data, as I describe later. It also allows us to interpret lt as potential
labor supply, and therefore an increase in unemployment would be seen as a fall in
productivity. Hence, in the current recession, model productivity is below trend,
whereas some measures of labor productivity in the data are above trend. This
assumption is also realistic, I believe, in the sense that consumers view increases
in unemployment as the economy using its resources inefficiently.
To construct the productivity series that the agent learns about, I take the real
per capita personal income series16 and regress ln yt = a0 + gt + εt . Growth in
the model is set to g and I calculate 
yt = ln yt − a0 − gt. I then choose zt so
yt )y ss .17 This ensures that income in the model
that model output ytmod = (1 + 
matches income in the data, allowing me to combine this model easily with the
credit constraint model.18
One way to set the prior beliefs would be to run a regression on the first few
quarters of data and set the priors on θ0 and P0 based on the regression estimates.
However, this approach leads to unstable estimates of the stationary model, where
productivity is predicted to go off to infinity. Instead, I take a different approach
and assume that the agent has some knowledge of the productivity process, even
though the quarterly data only begin in 1947.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

352

DANIEL L. TORTORICE
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1950

1960

1970

1980

1990

2000

2010

FIGURE 3. Probability weight on the nonstatationary model.

I set θ0s = [0.99 0 0 0] and make this prior as diffuse as possible while still
maintaining finite forecasts of productivity. This requires me to set P0S = 0.15×I.
I set θ0ns = [0 0 0] and set P0NS = P0S . I set σ02 = 0.012 (which is the full
sample estimate of the variance of the regression errors) and v0 = 1. To interpret
these priors, remember that (17) implies that the standard deviation of the initial
θ0 = 0.026. 19
Finally, I set k0 = k ss , ps,0 = 0.25, and lag = 4. Robustness to the choice of
initial lag and prior probability in shown in Section 7. Finally, to compare model
consumption to consumption in the data, I multiply model consumption by (1+g)t
and scale consumption by the mean of income in the data divided by the mean of
income in the model.
From the learning model, I generate three consumption series: one based on
learning about the parameters of the two income processes and the likelihood of
each model, one based on setting ps,t = 0 for all t (nonstationary model), and
one based on setting ps,t = 1 for all t (stationary model). For the latter two
models, I set θ S and θ NS to the O.L.S. estimates of the complete productivity time
series. They do not change over time. These last two series are my benchmark
no-learning models.
5.3. Dynamics of the Relative Variances of Consumption
and Income Growth
Figure 3 plots the agent’s probability weight on the nonstationary productivity
model implied by the learning model. Recall that the initial prior is 0.75 in 1947.
As is evident from the graph, the probability falls and then rises, reaching a peak in

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

353

1.2

1
Data
Learning
Nonstationary
Stationary

0.8

0.6

0.4

0.2

1950

1960

1970

1980

1990

2000

FIGURE 4. Ratio of standard deviation of consumption growth to income growth (models).

1962. It remains at that level for about 10 years and then falls, nonmonotonically,
putting more and more weight on the stationary model as time passes. In the last few
years the probability on the nonstationary model has risen substantially, as income
has not returned to trend. This probability pattern suggests that consumption
variance relative to income variance would rise early in the sample and fall later
on. We saw that this was in fact true in U.S. data (Figure 1), so now we examine
how well the model captures this pattern.
Figure 4 plots the predictions for the variance of consumption relative to income
from the models with and without learning. The no-learning, nonstationary-come
model predicts that the ratio should have no trend. It also overestimates the level
of the ratio. The no-learning, stationary model again predicts that this ratio should
be roughly constant and substantially underestimates its level, especially early in
the sample.
In contrast, the learning model, shown in the middle of Figure 4, does much
better than these benchmark models. It predicts a rise and fall in consumption
volatility. There are a clear rise and fall in consumption variance relative to
income variance, peaking at the same time as the peak in the data. However,
the model predicts too little variance at the peak of consumption volatility and
from 1975 to 1990. (It is possible to raise the overall variance of consumption
in the learning model by increasing the prior weight on the nonstationary model
and by increasing γ ; however, I prefer to remain with my calibration to show
that one gets good results with a very reasonable calibration. Perhaps one of
the reasons the model overestimates the speed at which consumption volatility
falls is that actual individuals learn more slowly than a Bayesian learner.) It is

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

354

DANIEL L. TORTORICE

Credit and Nonstationary Model
1
Data
M odel

0.8
0.6
0.4

1950

1960

1970

1980

1990

2000

Credit and Stationary Model
1

Data
M odel

0.8
0.6
0.4

1950

1960

1970

1980

1990

2000

FIGURE 5. Ratio of standard deviation of consumption growth to income growth (credit
models).

important to underscore that the learning model replicates the rise and fall in the
variance of consumption without using information on consumption data. It is
quite remarkable that the model replicates this distinctive consumption pattern
using probabilities calculated only from income data.
Another interesting feature of the model is that it replicates the rise in consumption volatility at the end of the sample, coinciding with the most recent recession.
Because income has been so far below trend for so long, the model puts extra
weight on the nonstationary model, and this increases consumption volatility.
It may be puzzling that the learning model line and the stationary model line
cross. Recall that the learning model is updating the parameters on the productivity
process every period, whereas the stationary model uses the full sample estimates.
Hence, when the learning model crosses the stationary model, it is putting most of
the weight on the stationary model, and its parameters for the stationary model imply less persistent deviations from trend than in the no-learning stationary model.
This explains why the two lines cross. Also, volatility in the nonstationary model
is quite high because the full sample estimates imply substantial autocorrelation
in productivity growth.
Figure 5 examines the ability of the credit constraints model with and without
learning to match this fact. The credit constraint model combined with the nonstationary model generates a slight increase in consumption volatility over time. We
see this because the volatility of consumption in the nonstationary model is higher
than that in the credit constraint model. As consumption becomes slightly less

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

355

1.1
Data
M odel

1
0.9
0.8
0.7
0.6
0.5
0.4

1950

1960

1970

1980

1990

2000

FIGURE 6. Ratio of standard deviation of consumption growth to income growth (credit +
learning model).

constrained, consumption volatility rises. When mixed with the stationary model,
the credit constraints model delivers a rise and fall in consumption volatility
consistent with the estimated rise and fall in credit constraints. But the magnitude
is small because the estimated change in credit constraints is small. Mixing the
credit constraints and the learning model (Figure 6) gives results similar to the
credit constraints with stationary consumption model. The only difference is that
the rise and fall of consumption volatility is slightly higher with the learning
model because the learning model generates a larger rise and fall in consumption.
Both models, though, are unable to generate the low volatility of consumption
at the end of the sample, because estimated credit constraints have not fallen
enough. Remember, these are 10-year averages, so the relevant measure of credit
constraints is that average over the next 10 years. And although credit constraints
were low in 2010, they were quite high on average from 2000–2010.
The last figure also highlights an important contribution of the learning model.
If loosening of credit constraints is an important contributor to the reduction in
consumption volatility, it is important to have a good model of consumption choice
absent credit constraints. The credit constraints model is sensitive to the choice
of income process. If the income process is stationary, the credit constraints
model gives the correct qualitative result. However, if the income process is
nonstationary, the reverse is true—consumption becomes more volatile as credit
constraints loosen. One can view the learning model as completing the credit
constraints model. It allows one not to impose an income process in the face of
uncertainty as to which process is correct, and to bring the model a bit closer to
the data.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

356

DANIEL L. TORTORICE

5.4. Time-Varying Response of Consumption to Income
In Section 3 I found that the correlation of consumption and income has fallen
over time. In the equation  ln ct = α + mt  ln yt + εt , mt fell over time (mt =
m0 + m1 t with m0 = 0.55 and m1 = −0.124). Table 1 reports the ability of the
different models to match this fact. We first see that the benchmark models without
time-varying credit constraints or learning (those with a known stationary or
nonstationary productivity process) cannot match this fact (m1 is not significantly
different from zero). Both the learning model and the credit constraints model
improve over the benchmark models; these models predict a fall in m over time.
For the learning model m0 = 0.7 and m1 = −0.073, and for the credit constraint
model m0 = 0.73 and m1 = −0.05. Adding the time-varying credit constraints to
the learning model gives similar results. In this case, m0 = 0.81 and m1 = −0.06.
All models predict a smaller fall than is in the data; however, the learning model
comes closest to matching the data.
5.5. Breaks in the Variance of Consumption
In Section 3 I measured breaks in the absolute deviation of consumption growth
from its mean. Table 2 presents the results from this estimate. The nondurables
series has one break at 1977:4. The means for each segment are 2.65 and 1.87. The
simulated consumption series from the learning model is estimated to have one
break occurring in the first quarter of 1982.20 Although a few years from the break
estimated in the data, it is within the estimated confidence interval for the break
date in the data. The respective means for each segment from the consumption
model with learning are 1.5 and 0.94. The model matches both the number of
breaks and the relative magnitude of the change. However, the break dates do
not line up exactly. None of the consumption measures from the benchmark nolearning models show evidence of any breaks. Consumption data from the model
with credit constraints show no breaks. When the credit constraints model is
combined with the learning model, there is one break in 1984, quarter 3, and the
mean falls from 1.77 to 1.21.
5.6. Relation between Consumption Changes and Probability Changes
The last few subsections argued that a learning model with changing probability
weights on the nonstationary model is consistent with the observed change in
the variance of consumption over time. This subsection extends that argument. In
the data, large changes in consumption are associated with large changes in the
estimated probability that the nonstationary model is true. I find that this fact is
replicated by the learning model.
To motivate this section’s analysis, imagine we want to investigate the hypothesis that changes in beliefs influence consumption choice. Then to test this theory,
we might want to identify the points in time that resulted in the largest changes

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

357

in beliefs and see if consumption changed more at those times than would be
predicted based on changes in income. Because my model identifies how much
beliefs should change at each point in time, I can use my estimated beliefs to
examine if changes in beliefs help explain consumption changes, controlling for
other factors that influence consumption. To this end, I estimate the following
regression on the consumption data:

ns 


  cs − ct−1

 + γ Xt + εt , (39)
 ln ct −  ln c = α + β ps,t − ps  ∗  t−1


ct−1
where the bar notation denotes the mean, ct is the NIPA measure of nondurables and
service consumption, ps,t is the estimated probability that the stationary model is
true (note that this is a function only of the income data and not of the consumption
data), and Xt is a vector of control variables, including a time variable21 and
cs −cns
| ln yt −  ln y|, where yt is personal income. | t−1ct−1 t−1 | is the absolute size of the
difference in the consumption choice implied by the learning model if the agent
puts probability one on the stationary model versus putting probability zero on the
stationary model, as a percentage of nondurables and service consumption. The
model implies that consumption changes are proportional to this term (see Section
6.2 for more details).
The results in Table 3 indicate that the absolute value of the probability change
is a significant predictor of the absolute value of the consumption deviation from
its mean. The coefficient β = 0.027. In other words, large changes in the estimated probability are associated with large changes in consumption controlling
for changes in income and time trends. In estimation I multiply both |ps,t − ps |
cs −cns
and | t−1ct−1 t−1 | by 100. Hence, this coefficient means that if consumption under the
stationary model differs by 10% and the probability changes by 0.01, the deviation
in consumption will increase by 0.27 percentage points, compared to a mean of
1.4 percentage points.
Table 3 reports the ability of the different models to replicate this fact. The
benchmark models without learning or time-varying credit constraints are unable
to replicate this fact. The model with learning naturally replicates this fact with
β = 0.067. However, the coefficient is too high. On the other hand, the model with
credit constraints predicts that β should be negative. Putting the credit constraints
model together with the learning model gets a coefficient that is close to correct,
β = 0.035.
6. MECHANISM OF THE LEARNING MODEL
To better explain the internal mechanisms of the learning model, I discuss two
implications of the model in more detail. First, I examine the largest movements in
the probability that the stationary model is true and show that they roughly accord
with intuition. When log productivity tends to be persistently above its steady
state value (0), we move toward the nonstationary model, and when productivity

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

358

DANIEL L. TORTORICE

TABLE 3. Effect of belief changes on consumption growth
Benchmark models
Data

Nonstationary Stationary Learning Credit

|(pst )−E[(ps)]|∗
0.027∗
|(cst−1 −cnst−1 )/ct−1 | [0.015]
0.24∗∗∗
|ln(yt )−E[ln(y)]|
[0.06]
t/100
−0.09
[0.14]
Constant
0.96∗∗∗
[0.21]
Obs.
221

0.004
[0.011]
1.01∗∗∗
[0.04]
0.02
[0.07]
0.15
[0.11]
221

Credit +
learning

−0.004
0.067∗∗∗ −0.019∗∗ 0.035∗∗∗
[0.006] [0.007] [0.009] [0.006]
0.46∗∗∗ 0.50∗∗∗ 0.65∗∗∗ 0.66∗∗∗
[0.02]
[0.03]
[0.03] [0.03]
−0.02
−0.18∗∗∗ −0.14∗∗∗ −0.16∗∗∗
[0.04]
[0.04]
[0.04] [0.04]
0.14∗∗
0.29∗∗∗ 0.33∗∗∗ 0.31∗∗∗
[0.06]
[0.08]
[0.09] [0.09]
221
221
221
221

Notes: Column (1) reports the results of regressing the absolute value of annualized consumption growth minus its
mean on the estimated change in the probability that the stationary model is true times the difference in consumption
predicted by the stationary and nonstationary models, controlling for the changes in income (y) and time trends.
Columns (2)–(6) redo the analysis of column (1) using simulated data from the nonstationary, no-learning model,
from the stationary, no-learning model, from the learning model, from the credit constraint model, and from the
credit constraint with learning model. Newey–West standard errors with five lags in brackets. ∗ Significant at 10%;
∗∗
Significant at 5%; ∗∗∗ Significant at 1%.

returns toward its steady state value, we move toward the stationary model. Second,
I decompose the consumption variance in the learning model into variance due
directly to shocks from productivity and that due to changes in the estimated
probability weights on each productivity model.
6.1. Largest Movements in the Productivity Process Probability
Recall, from (27), that the dynamics of the productivity process probability are
given by
	 NS 

wNS,t
 εt+1
wNS,t+1
	 S 

=
,
(40)
wS,t+1
 εt+1 wS,t


	
ti
	 i

 p θi , σi2 |Z
	 i 

i

,
	
= p yt+1
(41)
|xt+1
, θi , σi2
 εt+1
i
t+1
p θi , σi2 |Z
pS,t =

1
1+

wNS,t
wS,t

.

(42)

NS
S
)/(εt+1
) is lowest, and the five dates when
Table 4 lists the five dates when (εt+1
it is highest (and the data from the previous quarter), in order to build intuition for
the mechanisms of the model. These are the dates that provide the best evidence
for the stationary model and the nonstationary model, respectively. Additionally,
the table lists the percentage deviation of productivity from its steady state value

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

359

TABLE 4. Largest movements in the nonstationary probability
Year

Quarter

PNS

( NS )/( S )

Productivity

NS Forecast

S Forecast

2001
2001

1
2

Movements toward the stationary model
0.07
1.10
5.44
5.20
0.05
0.66
3.50
5.42

4.82
5.02

1974
1974

3
4

0.50
0.41

0.85
0.70

3.96
1.51

5.01
3.90

4.60
3.49

1949
1950

4
1

0.76
0.70

1.04
0.70

−8.83
0.74

−7.83
−8.82

−7.87
−7.76

2001
2001

2
3

0.05
0.03

0.66
0.72

3.50
1.72

5.42
3.42

5.02
3.08

1959
1959

1
2

0.55
0.47

0.89
0.73

−6.84
−4.88

−7.91
−6.83

−7.03
−5.89

2008
2009

4
1

Movements toward the nonstationary model
0.08
1.56
−9.02
−7.08
0.19
2.74
−13.45
−9.21

−6.80
−8.96

2008
2008

2
3

0.03
0.05

1.09
2.05

−3.83
−6.95

−3.35
−3.83

−3.10
−3.58

2009
2009

2
3

0.18
0.28

0.93
1.72

−13.28
−15.14

−13.73
−13.45

−13.37
−13.06

2004
2005

4
1

0.02
0.03

0.81
1.59

−2.81
−4.95

−4.18
−2.75

−3.92
−2.51

2008
2008

3
4

0.05
0.08

2.05
1.56

−6.95
−9.02

−3.83
−7.08

−3.58
−6.80

Note: This table lists the largest movements in the nonstationary probability along with the percentage deviation
of productivity from its steady state and its predictions.

(1) and the nonstationary and stationary forecasts of this productivity deviation
based on information from the previous period. At each date the stationary model
predicts that productivity from the previous period should move toward zero in the
next period. The nonstationary forecast always predicts that productivity should
be farther from zero than the stationary forecast predicts.22
The periods, then, that give the best evidence for the stationary model are
periods when productivity is far from zero and moves substantially toward zero.
In contrast, the periods that give the best evidence for the nonstationary model are
the periods where productivity is far from zero and moves even further away in
the next period.
For example, in 2001 quarter 2, productivity moves from 5.4% above trend to
3.5% above trend. This quick reversion to the mean generates evidence for the
stationary model. Similarly, during the latest recession, in quarter one of 2009

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

360

DANIEL L. TORTORICE

productivity falls from 9% below trend to 13% below trend, providing much
evidence for the nonstationary model.
This section helps illustrate that although this model may seem to be based
on a difficult econometric problem, too far detached from the decision making
of individuals, it can capture important features of actual consumption decisions.
First, imagine a situation such as the 1990s, in which the economy begins to
grow faster. Two views of the world may emerge. The first, the new era view,
assumes that the gains are permanent. A second, more pessimistic view would be
that the gains are temporary and the economy will eventually return to trend. If
uncertainty about these two views is important, when the recession of 2000 comes
along and gives evidence against the new era view, one can imagine an additional
effect on consumption from changes in beliefs, over and above the direct effect of
the productivity shock. I find support for this interpretation in Tortorice (2012).
This paper finds that past changes in unemployment affect the extent to which
individuals expect unemployment to mean revert. Second, note that the United
States has done comparatively well over the last 50 years, returning often to a
stable trend. One could imagine that, after seeing this repeated pattern, agents will
become more confident in the U.S. economy, and react less to recessions, seeing
them as temporary deviations from trend.

6.2. Decomposition of Consumption Variance
Figure 7 decomposes the variance of consumption growth relative to income
growth from the learning model into variance due directly to income shocks and
variance that comes from changes in the probability weight on each model. I
confirm numerically for the learning model that
ctl = ps,t cts + (1 − ps,t )ctns ,

(43)

where ctl is the consumption choice of the agent at time t from the learning
model, cts is the optimal consumption choice if ps,t = 1, and ctns is the optimal
consumption choice if ps,t = 0. This equation then implies that


	 s
ns
ctl = ps,t cts + (1 − ps,t )ctns + (ps,t − ps,t−1 ) ct−1
.
− ct−1

(44)

Therefore we can decompose the change in consumption into changes due
directly to income shocks and changes due to changes in beliefs:
ct = ct1 + ct2 ,
ct1

=

ct2 = (ps,t

(45)

+ (1 −


	 s
ns
.
− ps,t−1 ) ct−1
− ct−1

ps,t cts

ps,t )ctns ,

(46)
(47)

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

361

0.7
0.6
0.5
0.4
0.3
Model Variance
Variance from Shock
Variance from Belief Change

0.2
0.1
0

1950

1960

1970

1980

1990

2000

FIGURE 7. Variance decomposition—plot of the learning model’s prediction of
cov(ct1 , ln ct )
var(ln yt )

cov(ct2 , ln ct )
and its decomposition into
and var(ln
. Here ct1 =
yt )
cnst−1
s
pst ) × ct−1 , where cst is the consumption choice if pt = 1 and cnst
t−1 )
.
choice if pst = 0. ct2 = (pst − pst−1 ) × (cst−1c−cns
t−1

pst ×

cst
ct−1

var( ln ct )
var( ln yt )

+ (1 −

is the consumption

A standard variance decomposition gives


ct
Var
ct−1





ct1 ct
= Cov
,
ct−1 ct−1





ct2 ct
+ Cov
,
ct−1 ct−1


.

(48)

The first term, ct1 , represents the consumption response to a productivity
shock today. You are hit with a shock today, you put some weight on the shock
being permanent and some weight on it being temporary, and you adjust your
consumption accordingly. But because you revised your beliefs today, you also
believe that you made a mistake in your consumption choice last period. You put
too much weight on one of the models. And therefore you adjust your consumption
even more today, based on how much your revised beliefs indicate that last period’s
consumption choice was in error. This effect is captured by the ct2 term.
Figure 7 contains the results from this variance decomposition, normalized by
t
the variance of the income growth rate. Note that I report the variance of cc
t−1
23
predicted by the model, divided by the variance of income. There is a rise and
fall in the second covariance term early in the sample (marked x). This drives
about 15% of the rise in consumption volatility early in the sample. However,
by 1975, as the volatility of the probability changes has fallen substantially, most
of the variance of consumption is given directly by the response to changes

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

362

DANIEL L. TORTORICE
Lag 2 ps = .25

Lag 4 ps = .05

1

1

0.8

0.8

0.6

0.6

0.4

1960

1980

2000

0.4

Lag 4 ps = .25
1

0.8

0.8

0.6

0.6
1960

1980

2000

0.4

Lag 6 ps = .25
1

0.8

0.8

0.6

0.6
1960

1980

2000

1960

1980

2000

Lag 4 ps = .45

1

0.4

1980

Lag 4 ps = .25

1

0.4

1960

2000

0.4

1960

1980

2000

FIGURE 8. Robustness to different choices of lag length and prior: model (x); data (–).

in productivity, and the continued fall in the variance of consumption is due
to the increased weight put on the stationary model. Finally, during the most
recent recession, the weight on the nonstationary model rises dramatically and the
second covariance term again contributes about 15% of the observed volatility of
consumption.
7. ROBUSTNESS
Figure 8 explores the sensitivity of the results to different choices for the prior
probability. I plot the consumption standard deviation ratio from Figure 4 for the
learning model with an initial prior on the stationary model of 0.25 and for a lag
choice of 2, 4, and 6. Then I plot the same figure for priors on the stationary model
of 0.05, 0.25, and 0.45 with a lag length of 4.
As seen in column (1) of Figure 8, the initial rise in consumption variance
relative to income variance and the subsequent fall are present for all lag lengths.
Moreover, the fit gets better as the lag length increases; the fit is good for lags 4 and
6. The fit deteriorates slightly for a lag length of 2. The same rise and fall pattern
is evident, but the magnitude of the fall is muted. Similarly, the initial rise and
fall in consumption variance relative to income variance is visible for all priors.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

363

Decreasing the prior increases the mean level of the consumption variance and
increasing the prior decreases the mean variance, but the basic dynamic pattern of
a rise and fall in consumption volatility is not changed.
8. CONCLUSION
This paper studies three empirical observations. First, after an increase early in the
sample, the standard deviation of consumption growth relative to income growth
fell by 50%. Similarly, the correlation between consumption growth and personal
income growth has also fallen by about 50%. Finally, the consumption series has
one estimated break in the absolute deviation of consumption from its mean. The
change in mean deviation is about 30%.
I examine two explanations of these facts. The first is that a fall in the fraction
of credit-constrained consumers has lead to a reduction in consumption volatility
relative to income volatility. Although this model captured some of the fall in
consumption volatility, it was unable to match the full magnitude of the decline
in consumption volatility. It also left open the question of how an unconstrained
consumer would respond to an income shock, because if productivity shocks are
permanent, relaxing credit constraints could make consumption more volatile. To
address these shortcomings, I studied a model with an agent who learns whether
income shocks are permanent or transitory. This model was consistent with the
overall shape of the relative volatility of consumption to income, the declining
correlation of income and consumption over time, and the estimated break in the
consumption growth series.
The learning model in the paper has many applications. Perhaps one promising
route is to use learning about the specification of an exogenous process to add to
the literature that uses learning to explain large swings in asset prices [e.g., Bullard
(2010)].
NOTES
1. Section 6.1 provides more motivation as to why consumers share this econometric uncertainty.
2. As usual in the literature, consumption analysis is carried out using only data from 1955 onward,
avoiding the Korean War. Also, as Campbell and Mankiw (1990) point out, the National Service Life
Insurance (NSLI) benefits paid to WWII veterans in 1950:1 distort tests of the permanent income
hypothesis.
In theory 1950:1 could affect learning about the productivity process. To investigate this possibility,
I resimulated the model removing personal current transfer receipts, the component of income including
the insurance payment. Results were very similar. Because the 1950:1 observation does not influence
the results, I use all the income data to estimate the processes. Because beliefs about the productivity
process are central to this paper, I tie them to the data as much as possible.
3. I thank Kurt Kunze, an economist at the Bureau of Economic Analysis, for this insight.
4. Standard errors are calculated using the GMM delta method formulas. See, for example,
Cochrane (2001, 
pp. 207). In short, let μ = [E( ln ct )2 E( ln ct ) E( ln yt )2 E( ln yt )]
and φ(μ) =
E( ln ct )2 − [E( ln ct )]2 / E( ln yt )2 − [E( ln yt )]2 . Then var(φ) =
j =∞
1

[dφ/dμ]
xt−j )[dφ/dμ], where xt = [( ln ct )2 ( ln ct )( ln yt )2 ( ln yt )]. I
j =−∞ cov(xt ,
T
estimate the covariance matrix using a Newey–West estimator with 5 lags.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

364

DANIEL L. TORTORICE

5. Another approach, estimating mt with the Kalman filter, results in a similar conclusion. mt
declines approximately linearly from 0.44 to 0.21. I prefer this deterministic approach because I
need to make less restrictive assumptions about the nature of the error terms, and statistical inference
concerning mt is straightforward.
6. I chose not to include lag regressors in the regression equation that follows, so that the interpretation of the residual is clearer. I am looking to see if deviations in consumption about its mean have
changed in magnitude over time.
7. Estimated standard errors and confidence intervals allow for heterogenity and autocorrelation in
the residuals, using automatic bandwidth with AR(1) approximation and a quadratic kernel [Andrews
(1991)]. The residuals are AR(1) prewhitened and the variance–covariance matrix is allowed to vary
across segments. See Bai and Perron (2003) for details.
8. I use the Matlab code provided by Adda and Cooper (2003) to implement this method.
i=t
9. To calculate mpct , I simulate a series of ηi=0
and an alternative series that is the same except
p

 . Letting c = c |η


that ηt−1 = ηt−1
t
t t−1 , ct = ct |ηt−1 , yt = Et [yt |ηt−1 ], and yt
p

 ], the mpc =
Et [yt |ηt−1
t

ct −ct
p
p
yt −yt

=

.

10. The ex post real interest rate is measured as the three-month T-Bill rate minus the log change in
the GDP deflator.
11. The consumption change is measured in logs.
12. Modeling productivity as a deviation from an HP-filter trend would render productivity clearly
stationary, and the agent would learn that productivity is stationary very quickly.
13. Although the productivity process is nonstationary with some probability, the stability conditions
for the solution method still hold. See Section 4.2.3 for details.
14. The solution method requires that the eigenvalues of A − Bf , where the optimal policy is given
by u = −f x (where x is the state vector), are less than √1β in modulus. The computer program checks
this condition and finds it to hold. The matrices A and B are defined hereafter.
15. I take the approach of Kreps (1998) and Sargent (2001) of assuming that the agent treats its
probability weights and productivity process estimates as constants when calculating the optimal
policy. This “anticipated utility” approach is the most common in the learning literature and necessary
to have a tractable solution to the problem.
16. I use the personal income series instead of GDP because the empirical facts are based on the
personal income series.
1
17. zt = 1−α
[ln((1 + 
yt )y ss ) − α ln kt − (1 − α) ln(l ss )]. Recall that kt is chosen at time t − 1. To
initialize the model I need lag values of kt . I assume that kt = kss for t = 1, . . . , lag.
18. I have confirmed that the results for the learning model hold with more conventional measures
of productivity (e.g., TFP); however, the volatility of the income series in the model differs enough
from the income series in the data that when I combine the learning model with the credit constraints
model, it is unclear what income series to use to calculate, for example, fact 1, the standard deviation of consumption growth to income growth. I would get different results using income from
the learning model or using income from the credit constraints model that matches income in the
data. Similar issues would arise in analyzing the other facts as well. The current setup removes
this ambiguity, ensuring that the learning model generates income exactly equaling income in the
data.
19. I have taken care to make sure the results do not depend on the prior choice. Increasing P0NS to
1.5 × I did not affect the results, nor did reducing both P0S and P0NS to 0.015 × I , as long as I did
not update the A matrix for the first 10 periods to avoid unstable estimates of the stationary model.
Also, increasing θ ns = [0.05 0], lowering θ s = [0.95 0], and lowering P0NS to 0.015 × I caused no
noticeable change in the results except a larger increase in pns in the last few years. The results were
also insensitive to the choice of σ02 .
20. One can think of these breaks as “measured breaks” because the simulated data do not have
discreet breaks in consumption variance.
21. Higher-order time variables were not significant.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

365

22. Whether productivity is predicted to increase or decrease in the nonstationary model depends
on the lag values of zt and varies by observation.
23. This measure is not directly comparable to Figure 4, which plots the ratio of standard deviations;
however, the standard deviation, being nonlinear, does not decompose easily.

REFERENCES
Adda, Jerome and Russell W. Cooper (2003) Dynamic Economics: Quantitative Methods and Applications. Cambridge, MA: MIT Press.
Aguiar, Mark and Gita Gopinath (2007) Emerging market business cycles: The cycle is the trend.
Journal of Political Economy 115, 69–102.
Andrews, Donald W. K. 1991. Heteroskedasticity and autocorrelation consistent covariance matrix
estimation. Econometrica 59(3), 817–858.
Bai, Jushan and Pierre Perron (1998) Estimating and testing linear models with multiple structural
changes. Econometrica 66(1), 47–78.
Bai, Jushan and Pierre Perron (2003) Computation and analysis of multiple structural change models.
Journal of Applied Econometrics 18(1), 1–22.
Benito, Andrew and Haroon Mumtaz (2009) Consumption excess sensitivity, liquidity constraints, and
the collateral role of housing. Macroeconomic Dynamics 13(3), 305–326.
Bullard, James, George W. Evans, and Seppo Honkapohja (2010) A model of near-rational exuberance.
Macroeconomic Dynamics 14(2), 166–188.
Campbell, John Y. and N. Gregory Mankiw (1990) Permanent income, current income, and consumption. Journal of Business and Economic Statistics 8(3), 265–279.
Cecchetti, Stephen G., Alfonso Flores-Lagunes, and Stefan Krause (2005) Assessing the sources of
changes in the volatility of real growth. In Christopher Kent and David Norman (eds.), The Changing
Nature of the Business Cycle, pp. 115–138. Sydney: RBA Annual Conference Volume, Reserve Bank
of Australia.
Cecchetti, Stephen G., Alfonso Flores-Lagunes, and Stefan Krause (2006) Financial Development,
Consumption Smoothing, and the Reduced Volatility of Real Growth. AEA Conference Papers
2007.
Cochrane, John (2001) Asset Pricing. Princeton, NJ: Princeton University Press.
Cochrane, John H. (1988) How big is the random walk in GNP? Journal of Political Economy 96(5),
893–920.
Cogley, Timothy and Thomas J. Sargent (2005) The conquest of US inflation: Learning and robustness
to model uncertainty. Review of Economic Dynamics 8(2), 528–563.
Deaton, Angus (1992) Understanding Consumption. Oxford, UK: Oxford University Press.
Dynan, Karen E., Wendy Edelberg, and Michael Palumbo (2009) The effects of population againg on
the relationship among aggregate consumption, saving and income. American Economic Review:
Papers and Procedings 99(2), 380–386.
Dynan, Karen E., Douglas W. Elmendorf, and Daniel E. Sichel (2006) Can financial innovation help
to explain the reduced volatility of economic activity? Journal of Monetary Economics, 53(1),
123–150.
Evans, George and Seppo Honkapohja (2001) Learning and Expectations in Macroeconomics.
Princeton, NJ: Princeton University Press.
Georges, Christophre and John C. Wallace (2009) Learning dynamics and nonlinear misspecification
in an artificial financial market. Macroeconomic Dynamics 13(5), 625–655.
Guvenen, Fatih (2007) Learning your earning: Are labor income shocks really very persistent? American Economic Review 97(3), 687–712.
Hamilton, James (1994) Time Series Analysis. Princeton, NJ: Princeton University Press.
Kreps, David (1998) Anticipated utility and dynamic choice, a 1997 Schwartz Lecture. In Donald Jacobs, Ehud Kalai, and Morton Kamien (eds.), Frontiers of Research in Economic Theory,
pp. 242–274. Cambridge, UK: Cambridge University Press.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

366

DANIEL L. TORTORICE

Ljungovist, Lars and Thomas Sargent (2004) Recursive Macroeconomic Theory. Cambridge, MA:
MIT Press.
Ludvigson, Sydney (1999) Consumpiton and credit: A model of time-varying liquidity constraints.
Review of Economics and Statistics 81(3), 434–447.
Nakajima, Makoto (2007) Solving RBC Models with L–Q Approximation. Mimeo, UIUC.
Sargent, Thomas (2001) The Conquest of American Inflation. Princeton, NJ: Princeton University
Press.
Stock, James H. (1991) Confidence intervals for the largest autoregressive root in U.S. macroeconomic
time series. Journal of Monetary Economics 28(3), 435–459.
Stock, James H. and Mark Watson (2003) Has the business cycle changed and why? In Mark Gertler
and Kenneth Rogoff (eds.), NBER Macroeconomics Annual 2002, pp. 159–230. Cambridge, MA:
National Bureau of Economic Research.
Tauchen, George and Robert Hussey (1991) Quadrature-based methods for obtaining approximate
solutions to nonlinear asset pricing models. Econometrica 59(2), 371–396.
Tillman, Peter (2011) Parameter uncertainty and non-linear monetary policy rules. Macroeconomic
Dynamics 15(2), 184–200.
Tortorice, Daniel L. (2012) Unemployment expectations and the business cycle. B.E. Journal of
Macroeconomics (Topics) 12(1), 1–47.
Waters, George (2009) Learning, commitment and monetary policy. Macroeconomic Dynamics 13(4),
421–449.

APPENDIX: LEARNING MODEL
A.1. LQ APPROXIMATION
I approximate U (w) =

{[k α (ez l)1−α −i][1−(θl)φ ]}1−γ
1−γ

with w = [z k l i] about the steady state



w = [z k l i] . These calculations are based on Nakajima (2007) and Ljungovist and
Sargent (2004). We can take a second-order Taylor approximation as
1
U (w) ≈ U (w) + (w − w) J + (w − w) H (w − w) ,
2
where J is the Jacobian matrix and H is the Hessian matrix, both evaluated at the steady
state. This expression can be manipulated into this form:
⎡
U (w) − w J + 12 w H w
U (w) ≈ [1w ] ⎣
1
(J − H w)
2

1
(J
2

⎤
− H w)  1 
⎦
.
w
1
H
2

Let
⎡
U (w) − w J + 12 w H w
M=⎣
1
(J − H w)
2

1
(J
2

− H w)

⎤
⎦

1
H
2

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

CREDIT, LEARNING, CONSUMPTION VOLATILITY

367

and let x = [1 z k ] and u = [l i] . We partition M as

U (w) ≈ [1w ] [M]

1
w


=

  
 
x
R W
x
u
W Q
u

= x  Rx + u Qu + 2x  W u.

A.2. MODEL MATRICES
For the stationary model the state variables are xtS = [1 zt kt zt−1 . . . zt−p+1 ] and the control
variables are u = [lt it ] . So we have


R 0
s
R =
,
0 0
 
W
Ws =
,
0
Qs = Q.
The analogs for the nonstationary model are exactly the same.
For the nonstationary model the state variables are xtNS = [1 zt kt zt zt−1 . . .
i
=
zt−p+2 ] and the control variables are ut = [lt it ] . So for the laws of motion (xt+1
i i
i
i i
At xt + B ut + C εt+1 ), we have (for clarity I have written it for four lags, but it is easy to
generalize)
⎡
⎤
1 0
0
0
0
0
⎢
1
2
3
4 ⎥
⎢0 θs,t
⎥
0 θs,t
θs,t
θs,t
⎢
⎥
⎢
⎥
1−δ
0
0
0
0
0
⎢
⎥
S
1+g
At = ⎢
⎥,
⎢
⎥
⎢0 1
⎥
0
0
0
0
⎢
⎥
⎣0 0
0
1
0
0⎦
0 0
0
0
1
0

ANS
t

⎡
1
⎢
⎢0
⎢
⎢
⎢0
=⎢
⎢0
⎢
⎢
⎢
⎣0
0

0

0

0

0

1

0

1
θns,t

2
θns,t

0

1−δ
1+g

0

0

0

0

1
θns,t

2
θns,t

0
0

0
0

1
0

0
1

B s = B NS

⎡
0
⎢0
⎢
⎢0
=⎢
⎢0
⎢
⎣0
0

0

⎤

⎥
3 ⎥
θns,t
⎥
⎥
0 ⎥
⎥,
3 ⎥
θns,t
⎥
⎥
⎥
0 ⎦
0

⎤
0
0 ⎥
⎥
1 ⎥
1+g ⎥ ,
0 ⎥
⎥
0 ⎦
0

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417

368

DANIEL L. TORTORICE
⎡ ⎤
⎡ ⎤
0
0
⎢1⎥
⎢1⎥
⎢ ⎥
⎢ ⎥
⎢ 0 ⎥ NS ⎢ 0 ⎥
⎥
⎢ ⎥
Cs = ⎢
⎢0⎥ C = ⎢1⎥.
⎢ ⎥
⎢ ⎥
⎣0⎦
⎣0⎦
0
0

Note that the optimal policy is independent of C. Because the objective is linear–quadratic,
the optimal policy exhibits certainty equivalence.

Downloaded from http:/www.cambridge.org/core. University of Florida, on 25 Dec 2016 at 16:13:41, subject to the Cambridge Core terms of use,
available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S1365100512000417