BAYESIAN ESTIMATION OF THE N EUTRAL RATE WITH A STRUCTURAL O PEN -ECONOMY
MACROECONOMIC MODEL IN A MODERATELY R ICH DATA ENVIRONMENT 1
Paul Corrigan
Department of Monetary and Financial Analysis
Bank of Canada
234 Wellington Street
Ottawa, Ontario K1R 7S7, Canada
Abstract
I apply the techniques for Bayesian estimation of DSGE models in a moderately-data rich
environment suggested by Boivin and Giannoni (2005) to the estimation of a structural openeconomy model with Canadian and US data, and use the model to calculate short-run and longrun open-economy neutral rates. I find that the nominal and real neutral rates are estimated with
greater accuracy than found with other methods such as that of Laubach and Williams (2003) . I
also generate plausible point estimates of components of the Canadian neutral rate, including a
risk premium corresponding well with other evidence on Canada's risk profile.
1
I am grateful to Ian Christensen, Allan Crawford, Sharon Kozicki, Pierre St-Amant, Raphael Solomon, and Greg
Tkacz for helpful discussions and seminar participants at the Bank of Canada for useful comments. All remaining
errors are my own. Nothing in this paper is necessarily the opinion of the Bank of Canada. Send comments to me at
pcorrigan@bankofcanada.ca or phone +1-613-782-7948.
Introduction
The concept of the neutral rate of interest first proposed by Knut Wicksell, that is the rate
at which inflation will not accelerate or decelerate, has recently been re-popularized by such new
Keynesian authors as Woodford (2003), and motivated much research into estimating the
unobserved neutral rate. A popular method for estimating the neutral rate is that of Laubach and
Williams (2003) who tried a Kalman-filtering method similar to those tried to estimate the output
gap (e.g. Orphanides and van Norden 2002) and the natural rate of unemployment (e.g. Staiger,
Stock and Watson 1997). Laubach and Williams postulated an ad-hoc relation between the
output gap and the interest rate gap, along with a Phillips curve to identify the output gap, to
estimate a time-varying neutral rate. Unfortunately, estimation of the neutral rate with any degree
of precision using the Laubach-Williams approach is a formidable task Clark and Kozicki
(2004), as well as Laubach and Williams themselves discuss in detail the difficulties with
precisely estimating the real neutral rate.
A major flaw of the Laubach-Williams approach is a lack of enough theoretically
motivated structure to allow for much precision, except for estimates of the very long-run trend
of the neutral rate. Researchers interested in estimating neutral rates for open economies , in
particular, are short of theoretically consistent ways of accounting for changes in open-economy
variables in a Laubach-Williams style setup, be they short -run shocks to the current account or
longer-run changes in the risk profile of the economy in question.
One way forward with obvious appeal is use of a more structural model to calculate
estimates of the neutral rate. Neiss and Nelson (2001) and Lam and Tkacz (2004), tried to derive
the neutral rate as a function of “shocks” taken as the residuals of the first order conditions of the
equations of a calibrated model. Neiss and Nelson’s estimates of the “natural rate” are so volatile
as to make them of little use as a target. Lam and Tkacz’s “real interest rate gap” is much
smoother, and, moreover, does have information content for forecasting business cycles.
However, it is not clear a priori how sensitive a neutral rate estimated with a calibrated model
might be to changes in the structural parameters. (Lam and Tkacz only evaluate the sensitivity of
their real interest rate gap along a few dimensions.) The advances in estimation of dynamic
stochastic general equilibrium models in recent years (e.g. Smets and Wouters 2003) suggests
that it would be feasible, and more persuasive, to estimate any structural model one might use to
estimate the neutral rate. The advantage of a Bayesian approach over a standard maximum
likelihood approach to estimation is the ability to add prior information to the information
contained in the likelihood in a consistent fashion.
I estimated a New Keynesian IS-LM model with open-economy features similar to that of
McCallum and Nelson (2001), using Canadian data and US data (proxying for the world
economy) on output, employment, inflation, interest rates, and open-economy variables. As I
suspected more than one indicator might have information for a given theoretical variable such
and inflation, I estimated the model with a moderately-sized data set, distinguishing several
indicators for employment, inflation and interest rates for both Canada and the US, in the spirit
of the methods for Bayesian estimation of dynamic stochastic general equilibrium models with
large data sets proposed by Boivin and Giannoni (2005).
I used the estimated model to derive estimates of the real and nominal neutral rate for
Canada as functions of the underlying structural shocks. The model does generate reasonable
point estimates for both short-run and long-run concepts of the Canadian nominal neutral rate,
corresponding well to other evidence on Canada's monetary stance during the last twenty years,
and error bands narrow enough to make the estimates reasonably precise. As a by-product of the
estimation, I was able to generate plausible point estimates of the components of the neutral rate,
including the Canada/US risk premium. I found, as well, that imposing prior information from
other evidence on such components as trend inflation (available for Canada, whose central bank
has announced inflation targets) increased precision in estimation of the neutral rates and their
components, though not without cost in the plausibility of the point estimates.
Model
The open-economy model of McCallum and Nelson (2001), on which my model is based,
incorporates a demand for imports in the small open economy by putting imports into the
production function, as opposed to several other papers such as Gali and Monacelli (1999) where
demand for imports comes from putting imported consumer goods in consumer preferences.
McCallum and Nelson argue that their imports-in-production-function approach be tter matches
key properties of US data than an imports-in-utility-function approach. My model can be thought
of as a special case of the open-economy model used by Smets and Wouters (2002), which
allows imports to enter both the utility and production func tions.
In my model there are two countries. One is the small open economy or “local economy.”
The other, which I’ll call the “global economy,” might in fact the economy of the rest of the
world, or that of a trading partner that is large enough in comparison to the local economy that it
can be modelled as closed, and with which the local economy conducts all its trade. At any rate,
the global economy is to be regarded as block-exogenous to the local economy.
The worker-consumers in both economies have similar preferences in consumption and
leisure, but thanks to difference in institutions, the local and global economies may have
different levels of price stickiness and different monetary policies (in particular, differing
inflation targets). To produce goods for domestic consumption and possibly for export, local
firms use inputs of domestic labour and imported goods ; neither economy uses capital.
The local economy
Given logarithmic preferences (with habit formation) of local consumers, the expectations augmented IS curve of the local economy, written in terms of percentage deviations of local
consumption c L, t from a trend proportional to z L ,t τ L ,t (where z L, t is local technology levels and
τ L, t is the trend of local labour supply), deviations of local inflation π L from its current target
π L, t , and deviations of the nominal interest rate (1 + R )L ,t from its trend (viz. the product of the
inflation target π L, t , the long-run trend of the risk-free real rate ξ G ,t , and a local risk premium
ξ L ,t ), can be written
(1 +ˆ R)L ,t
= E t πˆ L ,t +1 +
1
(Et ∆zˆ L, t+1 + Et ∆τˆL ,t +1 + Et cˆ L, t+1 − cˆ L, t )
1−φ
φ
−
(∆zˆ L ,t + ∆τˆ L + cˆ L, t − cˆ L ,t −1 ) − Et bˆL ,t +1 + bˆ L, t − Et bˆG, t+1 + bˆG,t
1−φ
.
Here bL, t is a preferenc e shock specific to local consumers, while bG , t is a preference shock
affecting both the local and global economies.
I'll assume preferences to be linear in leisure, so that the local labour supply curve obeys
1
φ
φ
φ
cˆ L ,t −
cˆL ,t −1 +
∆zˆ L ,t +
∆zˆ L ,t = wˆ L, t ,
1− φ
1 −φ
1− φ
1− φ
where ŵt is the deviation of the wage rate ŵt from its trend, proportional to z L, t . (I will assume
that local workers cannot enter the global labour market, and vice versa.) It is easy to show that
the local IS curve can be re-written
(1 +ˆ R)L ,t = Et πˆ L,t+1 + (Et ∆zˆL ,t +1 + Et ∆τˆL,t+1 + Et wˆ L ,t +1 − wˆ L,t ) − Et bˆL ,t +1 + bˆL ,t − Et bˆG,t+1 + bˆG,t .
As I noted above, local firms use domestic labour h L and imports im to produce output
y L . I’ll assume a constant elasticity of substitution production function and write the production
function in terms of log deviations from trend so:
yˆ L, t = (1 − ω )hˆL ,t + ωimˆ t ,
where ω is the export-import share in the local economy. Goods can either be consumed at
home or expor ted, giving a market-clearing condition
yˆ L, t = (1 − ω )cˆ L, t + ω xˆ t .
Standard cost-minimization, along with the log-linear preferences of local consumers,
gives us labour demand and import demand functions obeying
wˆ L ,t = mcˆL ,t +
qˆt = mcˆ L, t +
( yˆ
( yˆ
− hˆL ,t
L ,t
)
σ
L, t
− imˆ L, t )
σ
,
where σ is the elasticity of substitution between labour and imports, and qt , the real exchange
rate (viz. the real purchasing power of global currency versus that of the local currency), is the
real price per unit for imports faced by local firms. It follows from the demand functions that
m cˆ L ,t = (1 − ω )wˆ L, t + ω qˆ t ,
that is, the marginal cost of local output mc L ,t is a function of the real exchange rate as well as
the output gap (or rather domestic consumption gap), with the weight of q t naturally being
determined by the weight of imports in the local production function.
As is usual in New Keynesian economies, the deviation of local inflation from its target is
a function of marginal cost. I’ll assume Calvo-style time-dependent price stickiness, as average
length of price stickiness is easy to calibrate or set priors for with reference to survey evidence
on intervals between price changes by a particular firm. That gives us
πˆ L ,t − Etπˆ L ,t +1 =
(1 − α L )(1 − α L β ) mcˆ
αL
L ,t
=
(1 − α L )(1 − α L β ) ((1 − ω )wˆ
αL
L, t
+ ωqˆ t ),
α L being the probability of a local firm’s being unable to adjust its price at time t.
The global economy
As I noted above, consumers in the global economy have preferences similar to those of
the local economy, so that the global expectation-augmented IS curve is
(1 +ˆ R)G, t = Etπˆ G, t+1 +
−
1
(Et ∆zˆ G, t+1 + Et ∆τˆG, t+1 + Et cˆG ,t +1 − cˆG, t )
1 −φ
φ
(∆zˆ G,t + ∆τˆG + cˆG, t − cˆG ,t −1 ) − Et bˆG,t +1 + bˆG ,t
1− φ
and the global labour supply curve obeys
1
φ
φ
φ
cˆ G,t −
cˆG ,t −1 +
∆zˆ G,t +
∆zˆ G, t = wˆ G, t
1− φ
1− φ
1− φ
1− φ
so that
(1 +ˆ R)G,t
= E t πˆ G ,t +1 + (Et ∆zˆ G , t +1 + E t ∆ τˆG ,t +1 + E t wˆ G ,t +1 − wˆ G , t ) − E t bˆG ,t +1 + bˆG , t
The main difference between the technology of the global and local economies is that the
weight of imports in the global production function is negligible. Hence I will assume firms in
the global economy use only labour in production, so that yˆ G ,t = hˆG , t . As the export share of the
global economy is negligible, the market clearing condition is yˆ G ,t = cˆG , t .
The global Phillips curve can be derived as just a special case of the local Phillips curve,
with the export-import share ω set to zero, so:
πˆ G , t − E t πˆ G , t +1 =
(1 − α G )(1 − α G β )
αG
mcˆ G ,t =
(1 − α G )(1 − α G β ) ˆ
wG ,t ,
αG
where α G is the probability a foreign firm’s price will be frozen at time t.
Uncovered interest rate parity and the open-economy Phillips curve
To link the local and global economies, we need to specify a process for the real exchange rate in
terms of fundamentals. The most theoretically pleasing (if not, I admit, the most empirically
watertight) way of doing this is to assume uncovered interest rate parity, so:
E t qˆ t +1 − qˆ t = rˆL ,t − rˆG , t ,
where rˆt = (1 +ˆ R)t − Etπˆ t+1 is the deviation from trend r of the expected real return on a
country’s bonds. Notice that UIP implies
E t qˆ t +1 − qˆ t = E t wˆ L ,t +1 − wˆ L ,t − E t bˆL ,t +1 + bˆL ,t − E t wˆ G , t +1 + wˆ G , t .
If for now we let X t = qˆ t − cˆL ,t + cˆG ,t + bˆL, t − bˆG ,t , we can write Et X t +1 = X t . The only bubble free solution to E t X t +1 = X t is X t = 0∀t . That gives us as a solution for the real exchange rate
qˆ t = wˆ L ,t − wˆ G ,t − bˆ L, t
giving us for the local open-economy Phillips curve
πˆ L ,t − Etπˆ L ,t +1 =
(1 − α L )(1 − α L β ) (wˆ
αL
L ,t
)
− ωwˆ G,t − ωbˆL ,t .
A shock resulting in a rise in the local rate of return, a fall in the global rate of return, or both,
will serve to lower the real exchange rate as local assets become more attractive, increasing
imports and local domestic consumption, as well as reducing local consumer prices.
Monetary policy and shock processes
To close the system I need to specify monetary rules for the two economies. I’ll assume
both follow interest rate rules in terms of deviation of inflation from target, so:
(1 +ˆ R )L ,t
= ρ r , L (1 +ˆ R )L ,t −1 + (1 − ρ r , L )(ρ π , L πˆ L,T ) + ε r , L ,t − ρ r , Lε π , L, t − ρ r , L ε ξ , G ,t − ρ r , L ε ξ , L, t
and
(1 +ˆ R )G,t
= ρ r ,G (1 +ˆ R )G , t −1 + (1 − ρ r ,G )(ρ π , G πˆ G ,T ) + ε r ,G , t − ρ r , G ε π ,G , t − ρ r , G ε ξ ,G , t .
As the inflation target is not a constant in either economy, there are two types of monetary shock:
a shock corresponding to a change in the inflation target, ε π , and a shock that does not, ε r . The
rules in both countries allow for some amount of interest-rate smoothing, captured in the
autoregressive term ρr . Note I will allow ρ r , L ≠ ρ r ,G and the weights on inflation in the two
rules to also differ, so that ρ π , L ≠ ρ π , G .
I’ll assume the local and global preferenc e shocks to be independent AR(1) processes, so:
bG ,t = ρ b,GGbG,t −1 + ε b,G, t and bL ,t = ρ b, LL bL ,t −1 + ε b, L, t .
This leaves us with the following system:
(1 +ˆ R)L ,t = Et πˆ L, t+1 + (Et ∆zˆ L ,t +1 + Et ∆τˆ L, t+1 + Et wˆ L ,t +1 − wˆ L, t ) − Et bˆL ,t +1 + bˆL ,t − Et bˆG, t+1 + bˆG,t ,
πˆ L ,t − Etπˆ L ,t +1 =
(1 − α L )(1 − α L β ) (wˆ
αL
L ,t
)
− ωwˆ G,t − ωbˆL ,t ,
(1 +ˆ R)L ,t = ρ r , L (1 +ˆ R)L ,t −1 + (1 − ρ r, L )(ρ π , L πˆ L,T ) + ε r, L ,t − ρ r , Lε π , L, t − ρ r, L ε ξ , G,t − ρr , L ε ξ , L, t ,
(1 +ˆ R)G,t
= E t πˆ G ,t +1 + (Et ∆zˆ G , t +1 + E t ∆ τˆG ,t +1 + E t wˆ G ,t +1 − wˆ G , t ) − E t bˆG ,t +1 + bˆG , t ,
πˆ G , t − E t πˆ G ,t +1 =
(1 − α G )(1 − α G β ) ˆ
αG
y G ,t ,
(1 +ˆ R)G,t = ρr ,G (1 +ˆ R)G,t −1 + (1 − ρ r, G )(ρπ ,Gπˆ G,T ) + ε r,G ,t − ρ r ,Gε π ,G ,t ,
bL, t = ρ b ,LL bL ,t −1 + ε b , L, t , bG ,t = ρ b ,GG bG ,t −1 + ε b ,G , t ,
which can easily be solved by standard methods for solving rational expectations models. With
the model solved, the labour demand and import demand functions
wˆ L ,t = mcˆ L, t +
(yˆ
L ,t
− hˆL ,t
σ
) and qˆ
t
= mcˆ L, t +
( yˆ
L, t
− imˆ L, t )
σ
,
can be used along with the analogous export demand equation, viz.
− qˆ t = mcˆG , t +
( yˆ
G, t
− xˆ L ,t )
σ
or xˆ L, t = yˆ G ,t + σ (wˆ G ,t + qˆ t ) ,
and the market-clearing conditions to solve for hˆL ,t and imˆ t .
To these I need to add the processes for the global inflation targets, the long-run trend of
the risk-free rate, the local risk premium, and the trends for local and global technology and local
and global labour supply. I will assume all of these to be random walks, technology and labour
supply having an upward drift; I allow shocks to global and local labour supply to be crosscorrelated. The resulting processes are:
ln π G, t = ln π G ,t −1 + ε π , G,t ;
ln z G , t = γ z ,G + ln z G ,t −1 + ε z ,G ,t , ln z L, t = γ z , L + ln z L, t −1 + ε z , L ,t , ;
ln τ G ,t = γ τ ,G + ln τ G ,t −1 + ε τ , G ,t , ln τ L ,t = γ τ , L + ln τ L, t −1 + ε τ , L ,t ,0 < corr (ετ , L ,t , ε τ ,G ,t ) < 1 ;
ln ξ G ,t = ln ξ G , t −1 + ε ξ , G ,t , ln ξ L, t = ln ξ L ,t −1 + ε ξ , L ,t .
It remains to specify the local inflation target. One way to specify the local inflation
target, analogous to that for the global inflation target, is as a random walk, with shocks to the
local target possibly correlated with shocks to the global target, to capture "global inflation"
shocks (cf. Ciccarelli and Mojon 2005), so that
ln π L, t = ln π L ,t −1 + ε π ,L ,t ,0 < corr (ε π , L, t , ε π , G , t ) < 1 .
A random-walk specification is appropriate enough (failing an obviously better one) for a
country like the US that does not have an announced inflation target, and so for which we must
estimate an implicit inflation target, having little prior information (outside the data) on the path
of trend inflation.
However, in the case of a country like Canada , whose central bank does have an
announced inflation target to which it presumably strictly adheres, and so for which we have
substantial prior information regarding the inflation target, such an approach is far from
satisfactory. Hence I also estimated (and report results below) for a version of the model where
the path of trend inflation follows a deterministic trend (which I will describe below)
corresponding to the path of the Bank of Canada's announced inflation target. 2 Such a
specification amounts, in Bayesian language, to imposing a degenerate prior on the value of
trend inflation after 1991. To be sure, assuming trend inflation to be known and equal to the
announced target is not completely satisfactory either, as the data potentially could be more
compatible in some periods with trend inflation being well away from the announced target.
The open-economy neutral rate
The ultimate objective of this exercise is to calculate a neutral rate for an open economy,
so I need now to define a neutral rate for the local economy. The neutral rate is often defined,
somewhat vaguely, as that rate on interest that prevails when the output gap is zero and inflation
is at trend. There are at least two possible interpretations of this definition. One is that the local
neutral rate is the rate compatible with a local monetary policy such that πˆ L, t = 0∀t . This neutral
rate I will call the "short-run neutral rate" for the rest of the paper.
Holding πˆ L, t = 0∀t implies marginal cost is at trend, so that
wˆ L ,t = ωwˆ G ,t + ωbˆL ,t ,
and so
2
This alternate specification was suggested by Allan Crawford.
(1 +ˆ R)L ( NEU , short), t = rˆL ( NEU ,short),t
(
= (1 − ω )(bˆ
)
= (1 − ω ) bˆL ,t − E t bˆL ,t +1 + bˆG , t − E t bˆG , t +1 + ω (E t wˆ G , t +1 − wˆ G ,t )
L ,t
)
− E t bˆL ,t +1 + bˆG ,t − E t bˆG ,t +1 + ωrˆG ,t .
In other words, the local neutral rate of interest is a weighted average of the prevailing global
real rate of interest and the neutral rate that would prevail were the local economy closed. To
maintain domestic inflation at target, the greater the degree of openness of a local economy,
monetary authorities must accommodate short-run changes in foreign real rates of return more,
and changes in local rates of return less, to avoid large changes in real exchange rates (with
resulting changes in the cost of production and so prices).
Another, long-run concept of the neutral rate, which I will call the "long-run neutral rate,"
is simply the long-run trend to which local interest rates will settle once all short run shocks that
could disturb output from trend and inflation from target have dissipated, so that
(1 +ˆ R ) L ( NEU ,long ),t = 0∀t . This is arguably closer to the concept of the neutral rates estimated by
Laubach and Williams, who modelled their neutral rate as a function of long-run factors such as
the rate of economic growth.
Notes on the data and estimation
To estimate the model, I used data on output, employment, inflation, interest rates and
open-economy variables from Canada (my local economy) and the US (proxying the global
economy). A natural starting place for the data set would be in 1976, where Canada's
employment series generally begin, but preliminary estimation results suggested that the 19791982 period was not well explained by the model when estimated with a data set including that
period. This is not surprising, given that the 1979-1982 period saw monetarist experiments in
Canada as well as the US which would not be well-modelled by the interest rate rules I'm
assuming that central banks follow. For the results I'll report below, I omitted the data from the
monetarist periods and used quarterly data for Canadian and US indicators running from 1986:I
to 2005:II.
In a similar fashion to Boivin and Giannoni’s (2005) case C, to measure real activity,
inflation and interest rates in each country, I use several data series that have a clear
interpretation in terms of the concepts in the underlying model, distinguished by iid measurement
error. Boivin and Giannoni argue that this measurement error may be particularly important in
individual inflation series. Theoretically, there is no limit to the number of real and nominal
series I could use in the estimation beyond that imposed by my ability to gather data. However,
Boivin and Giannoni's results suggest that there are diminishing returns to forecasting
performance from adding, particularly when there is a good chance that the implicit assumption
that measurement error is uncorrelated across series is being violated (that chance getting larger
with the data set, especially if the number of indicators of a single variable grows beyond a
certain point). So I used twenty-two series in estimation, described below, and no more than four
indicators for any given varia ble.
The Canadian real activity variables are the quarter-to-quarter changes in the logs of
seasonally adjusted real GDP, an hours index and total Canadian employment. The observation
equations are
∆ ln GDPCAN ,t = γ GDP,CAN + ∆ ln z L ,t + ∆ ln τ L ,t + hˆL ,t − hˆ L, t −1 + ν GDP, CAN ,t
∆ ln hCAN ,i , t = γ h ,CAN , i + ∆ ln τ L ,t + hˆL ,t − hˆL ,t −1 + ν h ,i ,t
where ν j is the mean-zero measurement error on variable j.
The inflation indicators for Canada are the change in the log of four inflation series (real
GDP deflator, CPI-all items, CPI-all items less food and energy, and CPI-all items minus eig ht
most volatile components), the observation equations being for each series
ln π CAN,i ,t = ln π L ,t + πˆ L ,t + ν CPI ,CAN,i ,t .
I assumed the short-run policy interest rate to be measured without error. The observation
equation for the short-term policy rate (taken to be Canada’s Bank rate 3) is:
ln (1 + BANK t / 400 ) = ln π L ,t + ln ξ L ,t + ln ξ G , t + (1 +ˆ R )L, t .
So as to mine information from the yield curve, I also included a measure of the long rate. A
simple arbitrage condition suggests the following observation equation for the long rate (yield on
10-year Governme nt of Canada bonds), which includes a risk premium term for the long rate,
taken to be constant:
ln (1 + GY 10 CAN ,t / 400 ) = rGY10− bank, CAN + ln π L, t +
1 39
∑ E t (1 +ˆ R )L ,t+ i . .
40 i = 0
The observation equations for the US series on real activity, inflation and interest rates
are substantially similar. The equations on cha nge in logged real GDP, a logged hours index,
logged payroll employment and logged household survey employment are
∆ ln GDPUS , t = γ GDP,US + ∆ ln z G , t + ∆ ln τ G , t + hˆG , t − hˆG , t −1 + ν GDP ,US ,t
∆ ln hUS ,i , t = γ h ,US ,i + ∆ ln τ G ,t + hˆG ,t − hˆG ,t −1 + ν h , G ,i , t
while the observation equations on three measures of inflation (change from four quarters
previous of CPI-all items, CPI less energy and CPI less food and energy) are
3
It is customary to use the overnight rate for the Canadian short-run interest rate; my use of the Bank rate is an
artefact of my preliminary estimation work where I used data from before 1975, when widely available data on the
overnight rate usually begins. Given that the overnight and bank rates did not differ appreciably in the period I'm
considering, I don't believe use of the overnight rate would appreciably affect the results.
∆ ln CPI US ,i ,t = ln π G ,t + πˆ G ,t + ν CPI , G ,i ,t .
I also use the US GDP deflator as an inflation indicator, allowing for the persistently lower trend
in the GDP deflator when compared to CPI, so that I add an additional term π ( GDPD−CPI ),US ,
assumed constant throughout the period, to the observation equation so that:
∆ ln GDPDUS , t = π ( GDPD−CPI ),US + ln π G ,t + πˆ G , t + ν GDPD,G ,i , t
The observation equations for the US policy interest rate (taken to be the Federal Funds
rate) and long rate (yield on 10-year Treasury bonds) are similar to those for Canada:
ln (1 + FFt / 400 ) = ln π G ,t + ln ξ G ,t + (1 +ˆ R )G , t
ln (1 + GY 10US ,t / 400 ) = rGY 10− FF ,US + ln π G ,t + ln ξ G ,t +
1 39
∑ E t (1 +ˆ R )G ,t+ i
40 i =0
It remains to specify the observation equations for the open economy variables. As there
is no investment or government spending in the model, domestic consumption corresponds to
final domestic demand, so I can express the ratio of Canadian final domestic demand to GDP as:
ln FDD CAN ,t − ln GDPCAN ,t = cˆ L, t − hˆ L, t + ν ( FDD / GDP ),CAN , t .
Finally, the observation equation for the differenced logged Canada/US nominal exchange rate is
∆ ln eCAN / US ,t = πˆ L ,t − πˆ G ,t + qˆ t − qˆ t −1 + ln π L, t − ln π G ,t + ν e ,t .
I estimated the model with the numerical Bayesian methods that have rapidly become
standard in estimating dynamic stochastic general equilibrium models (see, for example, Ahn
and Schorfheide 2005 for an overview). Solving the model given a set of values for the structural
parameters of the model gives us a state transition matrix, along with solutions for the
endogenous variables of the model as linear equations in terms of the states, which can be used
to construct the observation equations for the data. I then used the Kalman filter to construct a
likelihood for the data given the structural parameters, the trend growth rates of the observables
and the standard deviations of the shock variables and the measurement error terms; see
Hamilton (1994, chap 12) for details. Combining the likelihood with prior pdfs on the model
parameters, trend parameters and standard deviations gives us a posterior pdf which can then be
integrated by numerical methods (I used a standard random-walk Metropolis-Hastings algorithm)
to get posterior moments of interest. In particular, the Kalman filter makes it easy to calculate
posterior distributions filtered and smoothed estimates of the values of state variables in each
period, given a draw for the model parameters from the approximate posterior distribution, and
so to generate an unconditional posterior distribution for the values of the state variables in each
period. In this case, as the nominal and real neutral rates, short-run and long-run, are all functions
of the states, it is easy to calculate estimates of historical neutral rates. I report below posterior
distributions of the estimated smoothed historical values of the following variables of interest for
Canada from 1986:I to 2005:II:
•
the nominal short-run neutral interest rate
(
)
ln (1 + R )L ( NEU , short), t = ln π L ,t + ln ξ G , t + ln ξ L ,t + (1 − ω ) bˆL ,t − E t bˆL ,t +1 + bˆG , t − E t bˆG , t +1 + ωrˆG , t
•
the nominal long-run neutral interest rate ln (1 + R)L ( NEU ,long), t = ln π L, t + ln ξ G, t + ln ξ L ,t ;
•
the real short-run neutral interest rate
(
)
ln (1 + r )L ( NEU ,short),t = ln ξ G , t + ln ξ L ,t + (1 − ω ) bˆL ,t − E t bˆL ,t +1 + bˆG , t − E t bˆG , t +1 + ωrˆG , t ;
•
the real long-run neutral interest rate: ln (1 + r )L ( NEU ,long), t = ln ξG ,t + ln ξ L, t ;
•
the implicit inflation target ln π L ,t ; and
•
the implicit Canada/US risk premium ln ξ L ,t .
I do not have real prior information on more than a few structural parameters of the
model (none I could gain without peeking at the data before estimation, at any rate), and so I
used diffuse priors on structural parameters whenever feasible. The few informative priors on the
parameters of the structural model that I used for the results reported here are given in Table 1.
Survey data for the US reported by Blinder (1993) suggests that the average firm changes its
price every three quarters or so, so I chose prior pdfs on average price-stickiness in each
economy 1/ (1 − α L ) and 1/ (1 − α G ) such that average price stickiness in each economy is
between two and four quarters. As the ratio of imports to GDP is ω / (1 − ω ) , I picked a prior
allowing a range for ω / (1 − ω ) from 0.2 to 0.4.
(Table 1 about here)
On most of the other structural parameters I imposed relatively non-informative priors. In
particular, while it is customary in Bayesian applications to restrict the weights of inflation
deviations in the Taylor rule around 1.5, Taylor's (1993) original value, I decided to give uniform
priors to the reciprocals of the inflation weights in the local and global Taylor rules, ρ π , L
and ρ π ,G , restricting 1/ ρ π , L and 1 / ρ π , G to be between zero and one. Hence ρ π , L and ρ π ,G
themselves are restricted above one, ruling out indeterminacy, but their values are otherwise
data-determined. The smoothing parameters in the Taylor rules, ρ r , L and ρ r, G , were given
uniform pr iors restricting them between zero and one, as were the autoregressive terms for the
preference shock processes, ρ b, LL and ρ b, GG , and the rate of substitution between labour and
imports σ . I also imposed uniform priors between zero and one on the correlation coefficients
between Canadian and US labour supply shocks , corr(ε τ ,L ,t , ε τ ,G , t ) , and (in the stochastic local
inflation case) the correlation of inflation target shocks corr (ε π , L ,t , ε π ,G, t ) .
I also imposed diffuse priors on the standard deviations of the structural shocks and the
measurement error terms. The standard prior on a scale parameter that must be positive is one
proportional to its inverse, but in some cases this caused a pile-up problem towards zero. I was
wary of the practice of Boivin and Giannoni of imposing an informative prior on badly-behaved
standard-deviation terms to force them away from zero (as I have no real prior information on
their values), so I employed a few admittedly more ad hoc tricks of my own. My preferred tactic
was to impose a flat prior on the logarithm on the standard deviation term, which can take any
value on the real line, side-stepping the problem of the Metropolis-Hastings algorithm rejecting
too many draws as a result of hitting the zero constraint on the value of the standard deviation. In
the rare case where estimation problems persisted (for example, the variance of the log of a
standard deviation went to infinity), I fixed the term a priori to ze ro. The most important case of
this was the standard deviation term for the US technology shock ε z,G, t ; the posterior mode for
the term was very small, and I took this as evidence that technology shocks were, if not nonexistent, then small enough to be negligible in the US after 1986.
Finally, I imposed a fixed value of one on the discount parameter β equal to one for both
countries, approximating a value close to but not exactly one.
Notes on the deterministic trend for Canadian inflation
As I noted above, I estimated two versions of my model, one assuming the Canadian
inflation target follows a random walk, another assuming it is deterministic. I should explain just
how (and why) I specified the deterministic trend for the Canadian inflation target.
The Bank of Canada instituted inflation targeting in February 1991, announcing an
intermediate target of three percent CPI inflation to be achieved by December 1992, with trend
CPI inflation to be further reduced to two percent by December 1995. The Bank has three times
(most recently in 2001) renewed its agreement with the Government of Canada to maintain an
inflation target range of one to three percent. To make such a policy operational, the Bank
attempts to move inflation to the midpoint of the target range (two percent) over a six to eight
quarter horizon (for details, see, e.g., Bank of Canada 2001).
When specifying the deterministic path for trend inflation in the deterministic-localinflation specification, I interpreted the Bank's policy as follows. Target inflation is taken to be
three percent in 1991:I, two years (eight quarters) ahead of the period in which the three percent
target was supposed to be achieved (1992:IV). Between 1991:I and 1993:IV (again, eight
quarters ahead of 1995:IV, the period in which the two percent target was to be achieved), target
inflation follows a linear downward to two percent. In 1994:I and all periods after, target
inflation is held constant at two percent.
For periods before 1991, while no announced target is available for Canada, visual
inspection of the data suggests no evidence of large changes in the trend of inflation between
1986 and 1990. I decided to assume inflation between 1986:I and 1990:IV followed a constant
trend, which I allowed to be determined by the data.
Results and discussion
For each specification, I drew 90,000 draws from the Metropolis-Hastings algorithm,
discarding the first 10,000, leaving 80,000 draws from which I calculated the posterior moments
for the model parameters. To save computer time and memory, I only took a draw from the
posterior distribution of the smoothed estimates of the underlying state variables at each tenth
draw; so, the posterior distributions of the historical real and nomina l Canadian neutral rates,
implicit inflation target (when applicable) and risk premium, are taken from 8,000 draws. The
covariance matrix for the jumping distribution for the Metropolis-Hastings algorithm was the
inverse of the Hessian multiplied scaling factor tuned do as to permit an acceptance ratio of
about 0.25.
The posterior distributions of the structural parameters and the standard deviations of the
various shock processes are given in Table 2(a) for the stochastic-inflation-target specification.
Given my use of improper priors on many parameters, formal Bayesian model comparison is not
possible. However, if the model fits the data only under assumption of structural parameter
values that are far from what other evidence on their values suggest, it may mean the model is
badly specified along some dimensions. The posterior ranges for the price stickiness parameters
and the open economy shares in GDP are well in the tails of the prior distributions. According to
the model, average Canadian prices are stuck for no less than three and a half and perhaps as
long as five and a half quarters, while average US prices are allegedly stuck for anywhere from
four to six quarters; both these estimated ranges seem too high. Posterior estimates of average
price stickiness that are much higher than survey data suggests are reasonable are commonly
found in estimated Bayesian DSGE models, including the Smets and Wouters (2003) model, that
use Calvo-type price-stickiness. More puzzling is the posterior range of ω / (1 − ω ) between 0.32
and 0.6, most of that above the upper limit of 0.4 observed in the data in the early 2000's. As the
relation between US (real) interest rates and Canadian neutral rates is (in my model) related
to ω / (1 − ω ) , the relation between US and Canadian interest rates we actually see in the data is
much tighter than my model would allow for given observed import-GDP shares.
The posterior estimates of the two Taylor rules suggest very tight damping of deviations
of inflation from target, both in Canada and the US; the mean 1/ ρ π , L suggests a ρ π , L of about 8,
while the mean 1/ ρπ ,G suggests a ρπ ,G of about 6, well above Taylor's value of 1.5. This suggests
that movements in inflation in both countries were largely due more to changes in inflation
targets than in persistent deviations from otherwise constant or moving targets , in the US (which
did not have an explicit inflation target in this period) as well as in Canada. Monetary shocks,
permanent or temporary, seem to have been less important in the US than in Canada. A typical
temporary monetary shock in the US was on average less than one -fifth the size of a typical
temporary monetary shock in Canada, while a typica l US inflation shock was only three quarters
the size of a typical Canadian inflation shock. In both countries, however, monetary shocks were
fairly persistent, the "halflife" of a monetary shock being about eight months. We can also argue,
somewhat informally, that permanent inflation shocks are important in both economies by
pointing out that the hypothesis that the average size of inflation shocks is zero either in Canada
(with a t-statistic of 6.05) or the US (with a t-statistic of 7.84) must be rejected at any reasonable
significance level.
(Table 2(a) and 2(b) about here)
Table 2(b) gives the posterior mean and standard deviation for the specification in which
Canadian inflation is taken to have a deterministic trend. Most of the parameter estimates are
similar. The most important differences in the estimates of Canadian monetary policy are an
even lower mean 1/ ρ π , L , suggesting a ρ π , L of about twenty, and greater persistence in Canadian
monetary shocks, which now have a half-life of about a year. Such estimates are consistent with
the Bank's announced tight adherence to its announced target, and its announced policy of
allowing deviations from it to last no more than two years, as described above. Notice as well the
average size of a shock to the Canadian risk premium has grown by about a quarter; it will
become clearer below that some of the variation associated with shocks to the Canadian inflation
target has been re-assigned to the Canadian risk premium.
The model, rightly or wrongly, reconciles the wide, persistent swings in US policy rates
over the period alongside much smaller (persistent) deviations of inflation from trend by
attributing them to accommodation of large deviations in the US (global) neutral interest rate,
which, when the downward drift of the long-run US neutral rate are accounted for, have a halflife of about a year. (The most obvious example is the huge interest rate reduction of 2001,
which, to the surprise of many, did not raise inflation rates appreciably.) Canada -specific
preference shocks are much more persistent (with half-lives of nearly five years), even once
changes in the long-run risk-free rate and the Canadian risk premium are allowed for, reflecting
the persistence in Canadian trade balances.
This brings us to the main subject of interest, namely the model's estimate of the 95%
confidence band for the historical nominal neutral rate for Canada since 1986. The 95%
confidence bands for the stochastic inflation target specification are reported in Figur e 1(a) (in
black) , with the actual Bank rate plotted along with it for comparison (in red); those for the
deterministic target specification are reported in Figure 1(b). The narrow confidence band around
the neutral rate point estimates (about half a percentage point each way) is a great improvement
on the confidence intervals of more than two percentage points reported by Laubach and
Williams. Notice in particular how closely the turning points in the Canadian neutral rate match
the turning points in the Fed funds rate (in blue). As just noted, the model attributes most shortrun movements in the US policy rate to short-run movements in the global neutral rate, and so
short-run movements in the Canadian neutral rate should be expected to track short-run
movements in the Federal funds rate fairly closely.
(Figures 1(a) and 1(b) about here)
Figure 2(a) illustrates (in black) the 95% confidence band for the long-run neutral rate for
Canada, for the stochastic -target specification. Again, the difference between the neutral rate
estimates given by the stochastic inflation target specification and that given by the deterministic
inflation target specification (given in figure 2(b)) is small. In each case I have plotted the 10-
year Government of Canada bond rate (in blue), as well as the Bank rate (in red), alongside the
neutral rate. Notice how closely the long-run neutral rate tracks the 10-year bond rate; in my
model, temporary shocks almost completely dissipate after 10 years, so that the 10-year bond
rate very closely approximates ln π L ,t + ln ξ G ,t + ln ξ L, t .
(Figures 2(a) and 2(b) about here)
As a practical matter, the model suggests that the yield spread between the policy rate and
long-term bond yields is a good approximation of the deviation of monetary policy from neutral,
and so that yield spreads contain some information about how stimulative monetary policy will
be, at least in the long run. However, the confidence band around the long-run neutral rate is a
good bit wider than around the short-run neutral rate, though still narrower than Laubach and
Williams; the 95% confidence band suggests a margin of error of about a percentage point each
way, with a long-run neutral rate in mid-2005 of between two and four percent.
The nominal neutral rates for Canada, both short and long run, determined by the model
are fairly insensitive to the specification of the Canadian inflation target, stochastic or
deterministic. Where the specification of the Canadian inflation target makes a difference is in
the accuracy of the components of the nominal neutral rate, including the real neutral rate and (in
the Canadian case) the Canada/US risk premium. Adding prior information about one component
of the neutral rate (the inflation target) aids in identifying the other components.
(Figures 3(a) and 3(b) about here)
Figure 3(a) gives 95% confidence intervals for the real short -run Canadian neutral rates
under the assumption of a stochastic inflation target. The real short-run neutral rate's movements,
like those of the nominal, closely follow that of the federal funds rate. The estimates are fairly
precise in 2000's, with margins of error of about half a percentage point each way, but in the
early 1990's the margin of error can be as large as two percentage points, Imposing a
deterministic inflation target, however, tightens the estimates considerably, reducing the margin
of error in the mid 2000's to as low as plus or minus 0.25 percent, and in the early 1990's to not
much more to plus or minus one percent.
(Figures 4(a) and 4(b) about here)
Figures 4(a) and 4(b) give 95% confidence intervals for the real long-run Canadian
neutral rate under the two specifications. The point estimates of the long-run rates are reasonable,
peaking in the mid-1990's as Canada's fiscal situation deteriorates and falling afterwards in
response to the austerity measures of the federal government during the late 1990's. However,
the confidence bands are not much narrower than those found by Laubach and Williams for the
real long-run neutral rate; under the assumption of a stochastic inflation target, the margin of
error about 1.5 percentage points each way in the 2000's, so that in mid-2005 the real long-run
neutral rate was anywhere from zero to three percent and as high as two percentage points in the
early 1990's. Imposing the deterministic inflation target improves the accuracy of the long-run
neutral rate estimates considerably in the early 1990's, but only marginally in earlier and later
periods, the margin of error shrinking only to about plus of minus 1.25 percent.
To underline this point further, to suggest possible solutions to the problem, and to
illustrate the model's correspondence with other evidence on Canada's monetary stance and risk
profile as a way of checking specification, I have plotted (in figures 5 and 6) model estimates of
the implicit Canadian inflation target and Canadian risk premium. The estimates of the implicit
target are reasonable and not too imprecise; the 95% confidence band of the implicit inflation
target suggests a range of plus or minus one percent around the mean, much better than was
found by Leigh (2005) for the US. Particularly encouraging is the path of the point estimate of
the implicit inflation target, corresponding well to historical evidence on the Bank of Canada's
inflation target, falling from five percent in 1991 to two percent in 1996, remaining within a one
to three percent range thereafter.
(Figures 5(a) and 5(b) about here)
Figure 5(a) gives the confidence band for the Canada/US risk premium, given the
stochastic inflation target. The (mean) point estimates for are reasonable; a risk premium of
about half a percentage point rises in response to chronic federal deficits to around two percent
by 1995, before slowly declining in response to the austerity measures of the late 1990's back to
the range of about half a percent. However, the confidence bands are of a breadth comparable to
that of the inflation target, about a percentage point each way at the end of the period and as high
as two percent each way around 1990; such a range is clearly not as satisfactory for a risk
premium as it might for an inflation target range.
Imposition of the deterministic inflation target (Figure 5(b)) does not obviously improve
the risk premium estimates. As with the long-run real neutral rate, the deterministic inflation
target tightens the confidence bands of the risk premium in the early 1990's, but the improvement
is less dramatic in earlier and later periods. Also, the point estimate of the risk premium rises
around 2000-01, for reasons not obviously connected with changes in Canada's risk profile.
(Figures 6(a) and 6(b) about here)
To illustrate my own conjecture about why that might be happening, I have plotted the
implicit inflation target for Canada estimated by the model assuming a stochastic inflation target
in Figure 6(a). For comparison purposes, the inflation trend assumed by the model using a
deterministic inflation trend is given in Figure 6(b). With few exceptions, the point estimates of
the implicit inflation target since 1991 are within the target ranges to which the Bank was
committed in that period; in particular, in the period after 1996, except possibly around 1998, the
point estimate of the implicit inflation target is within the one to three percent target range to
which the Bank has been committed during that time. However, the movement of the implicit
target within that range is quite wide; notice in particular the rise of the point estimate to nearly
three percent around 2001-2002, a period in which CPI inflation, however measured, was on a
year-by-year basis nearer to three than two percent.
Long-run Canadian bond rates are, in my model, determined almost exclusively by the
Canadian inflation target, the risk-free (US) long-run real rate and the Canada/US risk premium.
If all movement in the inflation target is suppressed, any variation in the long Canadian bond rate
will be attributed to changes in the risk premium. If on the other hand the implicit inflation target
is allowed to "float," the data strongly favour changes in the long-run trend of inflation over
changes in the risk premium.
While the hypothesis that the "actual" target compatible with the Bank of Canada's policy
was completely constant appears to be rejected, I should add that the error bands around the
estimated targets are also fairly wide, up to plus or minus one percent in the mid-2000's. A
middle way between imposing the announced target a priori and a diffuse prior on the value of
the target at each point in time is estimation with informative priors on the time-path of the
target, shrinking the estimated inflation target towards a prior value (perhaps two percent) and
improving the precision of the estimates. I leave that to future work.
Conclusions
As a general rule, then, my model of the Canadian neutral rate in its current form does
quite well in precisely estimating (historical) short-run nominal and neutral rates to assess a
small open economy's policy stance, and the relationship of short-run neutral rates to global
interest rates. Encouraging as well are the model's point estimates of such constituent parts of
short-run and long-run neutral rates such as (particularly important for an open economy) risk
premia, which match well with other evidence with Canada's risk profile in the last twenty years.
As far as that goes, adding more structure to one's research, or at any rate one's estimates of the
neutral rate, does pay dividends.
However, the model in its current form is quite preliminary, and while the accuracy of
estimates constituent parts of the neutral rate of interest, such as the long-run trend of interest
rates and risk premia, is an improvement on that achieved by Laubach and Williams, more work
is needed in the future in estimation of the non-stationary elements of the neutral rate will be
needed before the model is useful for predicting neutral rates very far in the future. In future
research, I plan to explore how to exploit further sources of information that could pin down the
permanent determinants of neutral rates further. The most obvious way of doing this is to add a
richer variety of data to the model (for example, a larger array of interest rates to exploit more
yield curve information, as well as monetary aggregates and survey expectations of inflation to
pin down the target). I also plan to look at ways of adding information about state variable s (such
as Canadian inflation targets) with non-degenerate priors.
References
Ahn, S. and F. Schorfheide, 2005, Bayesian analysis of DSGE mode ls, Center for Economic
Policy Research Discussion Paper 5207.
Bank of Canada, 2001. Renewal of the inflation control target: background information. Ottawa,
Ontario: Bank of Canada.
Boivin, J. and M. Giannoni, 2005, DSGE models in a data-rich environment, manuscript,
Columbia University Department of Finance and Economics.
Blinder, A., 1994, On sticky prices: academic theories meet the real world, in: N. G. Mankiw,
ed., Monetary policy (University of Chicago Press, Chicago and London) 117-50.
Clark, T. and S. Kozicki, 2004, Estimating equilibrium real interest rates in real time, Federal
Reserve Bank of Kansas City Research Working Paper RWP 04-08.
Ciccarelli, M. and B. Mojon, 2005, Global inflation. European Central Bank Working Paper 537.
Gali, J. and T. Monacelli, 1999, Optimal monetary policy ande exchange rate volatility in a small
open economy, Boston College Working Paper in Economics 438.
Hamilton, J., 1994. Time series analysis (Princeton University Press, Princeton, New Jersey).
Lam, J.-P., and G. Tkacz, 2004, Estimating policy-neutral interest rates for Canada using a
dynamic stochastic general equilibrium framework, Bank of Canada Working Paper 2004-9.
Laubach, T. and J. Williams, 2003, Measuring the natural rate of interest, Review of Economics
and Statistics 85, 1063-70.
Leigh, D., 2005, Estimating the implicit inflation target: an application to U.S. monetary policy,
International Monetary Fund Working Paper 05/77.
McCallum, B. and E. Nelson, 2001. Monetary policy for an open economy: an alternative
framework with optimizing agents and sticky prices, Center for Economic Policy Research
Discussion Paper 2756.
Neiss, K. and E. Nelson, 2001, The real interest rate gap as an inflation indicator, Center for
Economic Policy Research Discussion Paper 2848.
Orphanides, A. and S. van Norden, 2002, The unreliability of output gap estimates in real time,
Review of Economics and Statistics 84, 569-83.
Smets, F. and G. Wouters, 2002, Openness, imperfect exchange rate pass-through and monetary
policy, Journal of Monetary Economics 49, 947-981.
Smets, F. and G. Wouters, 2003, An estimated dynamic stochastic general equilibrium model for
the euro area, Journal of the European Economic Association 1, 1123-75.
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unemployment? National Bureau of Economic Research Working Paper 5477.
Woodford, M., 2003, Interest and prices (Princeton University Press, Princeton, New Jersey).
Table 1: Informative prior pdfs on structural parameters
Parameter Distribution Mean Std. dev.
Gamma
2
0.5
α L / (1 − α L )
α G / (1 − α G )
Gamma
2
0.5
ω /(1 − ω )
Gamma
0.3
0.05
Table 2(a): Posterior distributions on selected structural parameters
(with stochastic Canadian inflation target)
Parameter
ρr , L
Monetary policy
Price stickiness
Open economy
technology
Preference parameters
and shocks
Supply shocks
Long-run
interest rate shocks
Mean (std. dev.)
0.7651 (0.0803)
1/ ρ π , L
0.1229
(0.0593)
σ ε ( r ,L )
0.3002
(0.0492)
σ ε ( π , L)
0.0829
(0.0137)
ρ r, G
0.7762
(0.0529)
1 / ρ π ,G
0.1598
(0.0529)
σ ε ( r ,G )
0.0552
(0.0107)
σ ε (π ,G )
0.0641
(0.0088)
corr (ε π , L ,t , ε π ,G , t ) 0.5472
(0.1505)
4.4088
(0.5250)
5.2099
0.4578
0.3594
0.9506
(0.5721)
(0.0704)
(0.1159)
(0.0104)
σ ε ( b, L )
2.2996
(0.4092)
ρ b, GG
0.8336
(0.0293)
σ ε ( b ,G )
1.4391
(0.2343)
φ
σ ε (τ , L )
0.7512
0.7224
(0.0288)
(0.0911)
σ ε (τ ,G )
0.5629
(0.0764)
0.3245
(0.1114)
σ ε ( ξ ,G )
0.1075
(0.0107)
σ ε ( ξ ,L )
0.0839
(0.0117)
1/ (1 − α L )
1/ (1 − α G )
ω /(1 − ω )
σ
ρ b, LL
corr (ε τ , L, t , ε τ ,G , t )
Table 2(a): Posterior distributions on selected structural parameters
(with deterministic Canadian inflation target)
Parameter
ρr , L
Monetary policy
Price stickiness
Open economy
technology
Preference parameters
and shocks
Supply shocks
Long-run
interest rate shocks
Mean (std. dev.)
0.8232 (0.0851)
1/ ρ π , L
0.0502
(0.0308)
σ ε ( r ,L )
0.3437
(0.0651)
π L (1986−1990)
1.0751
(0.0528)
ρ r, G
0.7676
(0.0487)
1 / ρ π ,G
0.1815
(0.0489)
σ ε ( r ,G )
0.0658
(0.0113)
σ ε (π ,G )
0.0564
(0.0076)
4.7963
(0.6112)
4.8456
0.4641
0.3580
0.9484
(0.6123)
(0.0685)
(0.1010)
(0.0100)
σ ε ( b, L )
2.2380
(0.3783)
ρb, GG
0.8231
(0.0342)
σ ε ( b ,G )
φ
1.3667
0.7517
(0.2413)
(0.0272)
σ ε (τ , L )
0.7643
(0.0949)
σ ε (τ ,G )
0.5522
(0.0702)
1/ (1 − α L )
1/ (1 − α G )
ω /(1 − ω )
σ
ρ b, LL
corr (ε τ , L, t , ε τ ,G , t ) 0.4772
(0.1187)
σ ε ( ξ ,G )
0.1069
(0.0094)
σ ε ( ξ ,L )
0.1012
(0.0112)
Figure 1(a): Mean (solid black) and 95% confidence interval (dotted black) of Canadian shortrun nominal neutral rate, compared to historical Bank rate (red) and Federal funds rate (blue),
1986:I to 2005:II, with stochastic Canadian inflation target
Figure 1(b): Mean (solid black) and 95% confidence interval (dotted black) of Canadian shortrun nominal neutral rate, compared to historical Bank rate (red) and Federal funds rate (blue),
1986:I to 2005:II, with deterministic Canadian inflation target
Figure 2(a): Mean (solid black) and 95% confidence interval (dotted black) of Canadian long-run
nominal neutral rate, compared to Bank rate (red) and 10-year Government of Canada bond rate
(blue), 1986:I to 2005:II, given stochastic Canadian inflation target
Figure 2(b): Mean (solid black) and 95% confidenc e interval (dotted black) of Canadian long-run
nominal neutral rate, compared to Bank rate (red) and 10-year Government of Canada bond rate
(blue), 1986:I to 2005:II, given deterministic Canadian inflation target
Figure 3(a): Mean (solid black) and 95% confidence interval (dotted black) of Canadian shortrun real neutral rate, 1986:I to 2005:II, given stochastic Canadian inflation target
Figure 3(b): Mean (solid black) and 95% confidence interval (dotted black) of Canadian shortrun real neutral rate, 1986:I to 2005:II, given deterministic Canadian inflation target
Figure 4(a): Mean (solid black) and 95% confidence interval (dotted black) of Canadian long-run
real neutral rate, 1986:I through 2005:II, given stochastic Canadian inflation target
Figure 4(b): Mean (solid black) and 95% confidence interval (dotted black) of Canadian long-run
real neutral rate, 1986:I through 2005:II, given deterministic Canadian inflation target
Figure 5(a): Mean (solid black) and 95% confidence interval (dotted black) of Canadian implicit
risk premium, 1986:I through 2005:II, given stochastic Canadian inflation target
Figure 5(b): Mean (solid black) and 95% confidence interval (dotted black) of Canadian implicit
risk premium, 1986:I through 2005:II, given deterministic Canadian inflation target
Figure 6(a): Mean (solid black) and 95% confidence interval (dotted black) of stochastic
Canadian implicit inflation target, 1986:I through 2005:II
Figure 6(b): Mean (solid black) and 95% confidence interval (dotted black) of deterministic
Canadian implicit inflation target, 1986:I through 2005:II