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DEBBIE GENDRON
Model stability under a policy shift:Are DSGE models really structural?
Mémoire présentéà la Faculté des études supérieures de l'Université Lavaldans le cadre du programme de maîtrise en économique
pour l'obtention du grade de maître es arts (M.A.)
FACULTÉ DES SCIENCES SOCIALESUNIVERSITÉ LAVAL
QUÉBEC
2007
©Debbie Gendron, 2007
Résumé
Ce mémoire développe un modèle dynamique stochastique d'équilibre général suivant lestraces de Ireland (2001), Dib (2003) et Dib, Gammoudi et Moran (2005). Le modèle estestimé à partir de séries chronologiques canadiennes via des techniques économétriquesbayesiennes qui consistent en la simulation des paramètres du modèle à partir de leurdensité a posteriori à l'aide de l'algorithme Metropolis-Hastings. Un test de stabilité dumodèle est ensuite effectué en comparant ses prévisions hors-échantillon lorsqu'il est es-timé avec deux échantillons différents. Le premier échantillon de données va de 1981T1à 2000T4, tandis que le second débute en 1991T1 afin de tenir compte de l'introductiondu régime de cible d'inflation par la Banque du Canada. Nous obtenons deux résultatimportants. Premièrement, les valeurs estimées des paramètres diffèrent de manière sig-nificatives selon que le premier ou le second échantillon est utilisé. Deuxièmement, la ca-pacité de prévision du modèle ne dépend pas de l'échantillon ayant servis à l'estimation.
Abstract
This thesis develops a dynamic stochastic gênerai equilibrium model (DSGE) in the lineof Ireland (2001), Dib (2003), and Dib, Gammoudi, and Moran (2005). The model is es-timated with Canadian time séries via Bayesian techniques by combining the likelihoodof its state-space représentation with prior information and simulating parameter valuesfrom their posterior density using the Metropolis-Hastings algorithm. A stability test isthen performed on the model by comparing its out-of-sample forecasting ability whenestimated on two différent samples. The first sample runs from 1981Q1 to 2000Q4,whereas the second starts in 1991Q1 to take into account the inflation targeting régimeintroduced by the Bank of Canada. Our main finding is that although the parameterestimâtes related to the monetary policy change significantly following the policy shift,the model's forecasting ability remains unaffected.
Acknowledgernents
When you've worked on a paper like this over a period of two years, in between classesand work, and full time during the summer, it's easy to forget how you started up, butyou'll never forget the people who encouraged and helped you throughout the journey.
The first two people I would like to thank are my thesis supervisors, Stephen Gordonand Kevin Moran, without whose excellent guidance the following would be nonexistent.
To Stephen I owe rny understanding of Bayesian econometrics, a branch of econo-metrics I was ignorant of upon my arrivai at Laval University. Stephen's patience andsupport combined with his ability to render some complex concepts so clear has beenof the greatest value. Also, I would like to thank him for his help with Matlab: withouthis preliminary programs, I would hâve been quite lost.
To Kevin I owe the discovery of macro models of the type in the following pages.His teachings hâve encouraged me to develop both criticism and curiosity as to what isout there to help with the understanding and analysis of macroeconomic interactionsand policy évaluation. I would also like to thank him for letting me knock on his doorwhenever I felt like it, for answering ail my emails and most trivial questions, and evenfor encouraging me to présent my paper in front of an audience.
To both and to the CIRPEE-Laval, thank you for the financial support that wasoffered to me, both in scholarships and as a research assistant.
To my family, thank you for encouraging me throughout my studies, and especiallyLyna who actually thinks I'rn the best!
To my friends, thanks to those who feigned interest in what I was doing and letme ramble on about what was right or wrong at the time. Thanks to those of youwho actually are interested and gave me input, asked questions or clarifications thathelped me better transmit what I had to say. Particularly, I would like to thank Luc
Acknowledgements
Bissonnette for his help on LATEX.
Also, I would like to thank ail the teachers I had but most especially ail the macroteachers, Pierre Fortin (UQAM), Victoria Miller (UQAM), Benoît Carrnichael (LAVAL)and Kevin Moran (LAVAL), for being so interesting and making me love macroeco-nomics.
To the reader, thank you for taking the time to read this thesis which represents asignificant part of rny éducation. I hope you will enjoy reading it as much as I enjoyedwriting and working on it.
Contents
Résumé ii
Abstract iii
Acknowledgements iv
Contents vi
List of Tables viii
List of Figures ix
1 Introduction 1
2 Models and macroeconomics 4
3 A policy régime shift in Canada 6
4 Model spécification 8
4.1 Households 84.2 The final-good-producing firm 114.3 The intermediate-good-producing firms 114.4 The monetary authority 134.5 Solving the model 14
5 Estimation and data 165.1 Bayesian econometrics: an overview 165.2 Bayesian estimation of a DSGE model 18
5.2.1 Priors 185.2.2 Likelihood 205.2.3 The Posterior density 21
5.3 Data 23
6 Estimation results 24
Contents vii
7 Forecasting comparison 27
8 Conclusion 34
Bibliography 35
A Symmetric equilibrium équations 39
B Transformed (stationary) System 41
C Steady-state solutions 43
D Linéarisation : Blanchard and Kahn 44
E Blanchard and Kahn's décomposition 47
F Using the Kalman filter for likelihood évaluation 51
G Posterior means and NSE 53
H Forecasting results 54
List of Tables
5.1 Prior distribution of parameters 20
6.1 Posterior Means of Parameters 26
7.1 Probability of the model's forecast 33
G.l Posterior means' point estimate 53
H.l Forecasting results using the entire sample period parameter's estimâtes 54H.2 Forecasting results using the subsample period parameter's estimâtes . 55
List of Figures
3.1 Canadian Time Séries 7
7.1 Forecasting Output 287.2 Forecasting Real Money Balances 297.3 Forecasting the Interest Rate 307.4 Forecasting the Inflation Rate 31
Chapter 1
Introduction
There is growing literature in macroeconomics using dynamic stochastic gênerai equi-librium (DSGE) models for analysis. Based on the gênerai equilibrium framework ofthe real business cycle (RBC) models, the DSGE models now routinely incorporatemarket imperfections and nominal rigidities and possess realistic business cycle prop-erties. The DSGE approach is to formulate a model as the resuit of intertemporal,optimizing behavior on the part of différent classes of économie agents. The dynamicproperties of the model are thus traceable to behavioral assumptions, leading to a highdegree of transparency. In contrast to purely statistical alternatives, DSGE models areless prone to instability because their structure is anchored in microeconomic theory.Consequently, they should be more robust to différent policy shifts. For this reason,they are increasingly seen as important policy analysis vehicles.1
The question we ask is: Are DSGE models really robust to policy shifts? In particu-lar, are they stable over time? An answer to thèse questions is necessary if DSGE modelsare to be used for policy analysis in an ever-changing environment. According to Lucas(1976), the décision from either the monetary or fiscal authority of a country to changea key économie variable in a world where agents are forward-looking and rational mustbe reflected endogenoùsly through the agent's optimal behavior. Authors hâve shown,however, that in some forward-looking models, structural parameters' estimâtes may beunstable over time. For instance, Ireland (2001) estimated the parameters of a DSGEmodel of the American economy over différent sample periods and showed that manystructural parameter estimâtes changed over subsamples. In addition, Estrella andFurher (1999) undertake stability tests in both forward-looking and backward-looking
1Por instance, the Bank of Canada lias recently published its technical report describing ToTEM - asticky-price dynamic stochastic gênerai equilibrium model of the Canadian economy - that has becomethe new quarterly projection model. http://www.bankofcanada.ca/fr/res/tr/trlist-f.html
Chapter 1. Introduction 2
models and find that some backward-looking models perforai better than their forward-looking counterparts as to parameter stability. Taking for granted that parameters in aforward-looking model are invariant could therefore resuit in very important forecastingor policy évaluation mistakes.
This thesis examines this important question of stability by estimating the structuralparameters of a DSGE model with Canadian data, first using a sample of data runningthrough 1981Q1-2000Q4, and then repeating the exercise with a subsample starting in1991Q1. Afterward, we compare the out-of-sample forecasts of the model estimated onthe différent samples. We chose 1991Q1 as a breaking date because in 1991, the Bank ofCanada started inflation targeting which constitutes an important shift in the practiceof monetary policy in Canada.2 Estimation of the model's parameters is conducted witha Bayesian approach (e.g. following Smets and Wouters, 2003), and the stability test isconducted by comparing the out-of-sample forecasts of the model estimated under thetwo différent samples with the use of Bayesian model comparison.
Our main finding is that although some parameters significantly change followingthe policy shift, the model's forecasting ability remains unaffected. The only parametersthat do change are those in the monetary policy rule, which indicates that the model hasbeen able to capture the policy shift of 1991. As to the other structural parameters,they remain stable over time. When comparing the forecasting ability of the modelestimated over our two samples, neither method overwhelmingly outperforms the other.This allows us to conclude that the model is stable over time.
To our knowledge, this thesis is the first attempt to test for parameter stability ina DSGE model of the Canadian economy. In addition, although Canadian data hasbeen used before to estimate DSGE models (Dib 2003, Dib, Gammoudi and Moran2005), this thesis reports the first Bayesian estimation of a DSGE model in a Canadiancontext. The use of a Bayesian approach is very interesting in this context and allows usto explicitly use prior information about parameter values which renders unnecessarycalibration of certain parameters and therefore yields a complète description of ailparameters' posterior distribution.
The remaining of this thesis is organized as follows: Chapter 2 takes a closer lookat the Lucas Critique and the évolution of macroeconomics leading up to the DSGEmodels. In Chapter 3, we propose évidence of a policy shift in Canada and its im-portance on the data. Chapter 4 develops the prototypical DSGE model under study.In Chapter 5, we describe the estimation methodology of the model's parameters andthe data. In Chapter 6, we expose our estimation results and compare them to other
2Bordo and Redish, 2005.
Chapter 1. Introduction 3
studies. We perform the forecasting test and discuss our results in Chapter 7. Finally,Chapter 8 will conclude the thesis.
Chapter 2
Models and macroeconomics
Let us begin with Robert Lucas' séminal paper: "Econometric Policy Evaluation: ACritique" (1976). In this article, Lucas criticizes 1960s-style macroeconomics in whichthe economy was modeled as a System of équations based on décision rules. The idea Lu-cas puts forth is that behind thèse décision rules lies the behavior of différent économieagents, and the behavior of thèse agents is determined through expectations about thefuture state of the economy. Estimating the parameters of the décision rules, and thentaking them as given in order to conduct policy évaluation can therefore be misleadingif agents adjust their behavior to policy changes, thus modifying the underlying pa-rameter values of the équations. Lucas points out that macroeconomic models shouldbe developed at the level of microeconomic décisions and insists on the importance ofmodeling agents' behavior.
To answer Lucas' criticism, a new methodology is developed in the early 80s, par-ticularly by Kydland and Prescott (1982), known as the real business cycle (RBC)methodology. In thèse models, explicit intertemporal optimization problems are pre-sented and solved, and gênerai equilibrium is the aggregation of optimal behavior in thedifférent sectors of an economy. With this framework, dynamic behavior is traceableto tastes and technologies of the différent decision-making agents in the economy andtheir best expectations regarding the future state of the economy. Thus, the structureof the economy is detailed in a way that is cohérent with Lucas' critique.
However, the canonical RBC rnodel is soon subjected to criticism because it is unableto replicate certain aspects of the observed business cycle.1 The criticism however isnot directed to the modeling framework, but rather toward certain assumptions of
xFor example, Cogley and Nason (1995) show that many RBC models hâve weak internai propaga-tion mechanisms.
Chapter 2. Models and macroeconomics
the standard model. For instance, business cycle fluctuations in an RBC model arisefrom technology shocks only, while many researchers believe that additional shocks areresponsible for some of the aggregate fluctuations.2 In particular, the earlier modelsare silent about the rôle of monetary policy in business cycle fluctuations even thoughthere is ample évidence that "Money Matters"3 and that the short-run stabilizationcapabilities of the monetary authority can only be modeled in an environment withsome type of nominal friction, absent in the RBC model.
The more récent stages of this branch of macroeconomic modeling, known by itsacronym DSGE (dynamic stochastic gênerai equilibrium) models build on the basicRBC model by adding features better equipped to replicate business cycle fluctuationsand to address monetary policy issues. Many of today's researchers interested either inpolicy évaluation or forecasting for example hâve been using the DSGE framework fortheir analysis. The most common models incorporate either price or wage rigidities, orboth, and an interest-rate rule followed by the monetary authority. Depending on theinterests of the researchers, some will work on an open-economy model, while otherson a closed-economy model, some will add a government sector, etc.4 But even thesimplest models offer very encouraging results both in their forecasting performanceor their ability to match business cycle properties. Consequently, they hâve becomevery attractive alternatives to more statistical models as policy analysis tools for bothmonetary and fiscal authorities.
2See Smets and Wouters (2003) for an example of many différent shocks that may be présent in aneconomy.
3See Romer and Romer (1999).4We refer the reader to Smets and Wouters (2003) and Ratto, Roger, Veld, and Girardi (2005) for
treatment of the European area using a DSGE model; to Ireland (2001), Smets and Wouters (2005)and Ambler, Guay, and Phaneuf (2003) for treatment of the US economy; to Dib, Gammoudi, andMoran (2005), Dib (2003) and Amano and Murchison (2005) for examples using Canadian data. Thèseare but a few of the existing literature on DSGE models.
Chapter 3
A policy régime shift in Canada
The Bank of Canada is Canada's central bank and is responsible for the control ofmonetary policy, with an aim to promote the économie and financial well-being ofCanada. Over the past thirty years, there hâve been différent views taken by the Bankas to what économie indicators it should use to justify action on financial markets.Prom 1975 to 1982, the Bank's immédiate target was the short-term interest rate andits intermédiare target was Ml. Because of the strong demand-elasticity of Ml tothe short-term interest rate, the variations in the interest rate necessary to respectthe targeted expansion of Ml were too small to prevent large gaps in total spendingand inflation compared to their desired trajectories. Ml was thus abandoned as atarget in 1982. The following décade seems to hâve been occupied by the search ofa new monetary aggregate to replace Ml. The absence of a consensus led the Bankto concentrate on its monetary policy ultimate objective: price stability (Duguay andPoloz, 1994).
Following this décade of uncertainty, an inflation target was jointly agreed upon in1991 by the Canadian government and the Bank of Canada. The initial goal was toreduce inflation progressively so the inflation target followed a declining schedule: firstto 3 per cent, then to 2.5 per cent, finally to 2 per cent to ensure a climate favorable forlong-lasting économie growth. By December 1993, inflation had been reduced to 2 percent. At that time, the government and the Bank agreed to extend the inflation-controltarget range until the end of 1998, with a range of 1 to 3 per cent. In February 1998,the target range was extended to the end of 2001, in May 2001, it was renewed untilthe end of 2006, and on November 23 2006, it was again renewed until 2011.
Figure 3.1 depicts the time séries of important Canadian macroeconomic variablesover the sample period we are interested in (1981-2005). The data is taken from Statistic
Chapter 3. A policy régime shift in Canada 7
Canada's database (see section 5.3 for a detailed discussion of the séries in question).Notice that there seems to be a change in the évolution of the time séries around thebeginning of inflation targeting in 1991. Output shows a swift décline, followed bya period of stagnation and then growth, while real money balances enter a period ofreduced growth, whereas inflation and interest rates are both on a declining path. Inaddition, there appears to be a decrease in the variability of output. This impression isconfirmed by Debs (2001). He shows that there was a structural break in the variabilityof Canadian production growth during the first quarter of 1991.
The introduction of inflation targeting appears to hâve lessened uncertainty associ-ated with inflation as credibility of the Bank's intentions has grown and has stabilizedexpectations. Together, less uncertainty, and more stable expectations hâve led to anincrease in the length of différent contracts both on fmancial and labor markets.1
The introduction of inflation targeting in 1991 therefore constitutes an importantshift in the practice of monetary policy in Canada, which may hâve significantly mod-ified the behavior of différent Canadian time séries. In this context, it is important toestablish whether this shift in monetary policy has had an impact on macroeconomicmodels. In particular, are DSGE models such as described in the previous sectionimmune to such a shift? Are parameter estimâtes modified following the 1991 policychange? And if so, are thèse changes important for the forecasting accuracy of themodel? Thèse are the questions we are answering in the following sections.
Figure 3.1: Canadian Time Séries
10.9
1(1.11
10.7
I0.6
10.5
10.4
Log GDP per capita
yLog Real Money Balances per capita
1980 1985 1990 1995 2000 2005 2010
Annualized Interest rate
1980 1985 1990 1995 2000 2005 2010
Annualized Inflation rate
1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010
1See Longworth (2002) for évidence of lengthened contracts.
Chapter 4
Model spécification
In this chapter we formulate the DSGE model estimated in Chapter 5. The model understudy is a closed economy model of the Canadian economy similar to the one presentedin Dib, Gammoudi and Moran (2005), Dib (2003), and Ireland (2001). The economyis populated by households, final-good-producing firms, intermediate-good-producingiîrms, and a monetary authority. The final good market is perfectly compétitive: eachfirm produces the saine final good and is a price taker for input and output priées. Theprice of the final good adjusts to clear the market. Final good production is dividedbetween consumption and investment. The extent to which the capital stock can bemodified through investment is restricted by capital adjustment costs borne by thehouseholds who own the economy's capital. Final good production uses as inputs anarray of intermediate goods. In contrast to the final good's market, the intermediategoods' market is monopolistically compétitive. Taking as given the demand arisingfrom final-good production, each intermediate-good-producing firm produces a distinctgood, under the constraint that rigidities (following Calvo, 1983) affect its ability tochange the price of its good. Finally, the monetary authority responds to changes ininflation, output, and money growth by managing a short-term nominal interest rate,similar to Taylor (1993).
4.1 Households
The représentative household has préférences defined over consumption Ct, real moneybalances Mt/Pt, and leisure l-Ht (where Ht represents hours worked) during each periodt = 0,1, 2... as described by the following expected lifetime utility function:
Chapter 4. Model spécification
(4.1)
where j3 G (0,1) is the discount factor. The period utility function is specified as:
where 7/ and 7 are positive structural parameters. The first governs the disutility of
labour (utility of leisure), and the second is related to the intratemporal elasticity
between consumption and real money holdings. zt and bt are two serially correlated
shocks. The préférence shock zt, as shown by McCallum and Nelson (1999), resembles
a shock to the IS curve in more traditional Keynesian analysis. On the other hand, bt
is interpreted as a shock to money demand. The two shocks evolve according to the
following autoregressive processes:
log(zt) = pzlog(zt-i) + ezt, (4.3)
log(bt) = (1 - pb)log(b) + pblog(bt-i) + ebt, (4.4)
with pz and pb G (—1,1), and tzt and tu are serially uncorrelated innovations that are
normally distributed with zéro mean and standard déviations oz and ab, respectively.
The représentative household carries Mt-i units of nominal money balances, Bt-i
units of bonds, and Kt units of capital into period t. During period t, the household re-
ceives a lump-sum nominal transfer Tt from the monetary authority as well as dividend
payments Dt from the intermediate-good-producing firms. Further, the household sup-
plies labor and capital to the intermediate-good-producing firms, for which it receives
total factor payment RktKt + WtHt , where Rkt is the nominal rental rate for capital,
and Wt is the nominal wage. The household uses some of its funds to purchase the
final good at the nominal price Pt, which is then divided between consumption and
investment. The remainder of available funds are allocated to money holdings Mt and
financial bonds Bt, which are priced l/Rt, where Rt dénotes the gross nominal interest
rate between t and t + 1. The household's budget constraint is therefore given by:
Pt(Ct + h) + Mt + ^ < RktKt + WtHt + Mt-i + Bt_i + Tt + Dt. (4.5)
Investment It increases the capital stock, Kt, over time according to:
2
Kt+1 = (1 - 6)Kt + h - | ( ^ - g) Ku (4.6)
xThe utility fonction as formulated hère is a CES between consumption and real-money balances,and separable with leisure. Another formulation that has recently been growing in popularity is thatof habit formation. However, as demonstrated in CEE (2005), habit formation plays a central rôleonly when matching consumption, a dimension of the data that isn't the focus of our analysis.
Chapter 4. Model spécification 10
where ô E (0,1) is a constant capital dépréciation rate, y? > 0 is the capital adjustmentcost parameter, and g is the steady state growth rate of the economy to be discussedfurther below. With this configuration, the cost of changing the capital stock increaseswith the speed of desired adjustment giving the household an incentive to change capitalinvestment gradually. In addition, total and marginal capital adjustment costs equalzéro in the balanced-growth steady state.
The représentative household chooses Ct, Mt, Ht, Kt+i, Bt in order to maximizeexpected lifetime utility (4.1). The problem can be written in its recursive form, withits optimal solution satisfying the following Bellman équation:
V(Kt,Bt^,Mt^Qt)= max \U(ct, ,Ht) + EtpV(Kt+l,Bt,Mt;nt+l)] ,{ C t , M t , H t , B t , K t + \ \ L i t J
(4.7)where Q.t is the information available at time t. The décision making process of house-holds must respect conditions (4.5) and (4.6).
Solving for the représentative household's optimal behavior yields the following firstorder conditions:
Ct •• ^ E ï F ~ - A" (4-8)<V +W(Mt/Pty
'y
- i. ztW(Mt/Pt)~ PtXt+1\
Mt . —J=J j — - At ~ pk,t — . (4.9)
(4.10)
1 gPt+i *\Kt+1
yJKt+1 \ t +
(
where À is the Lagrangian multiplier associated to the budget constraint (4.5).
As in Ireland (1997), and Dib, Gammoudi, and Moran (2005), équations (4.8), (4.9),and (4.12) together imply the following standard money-demand équation:
log{Mt/Pt) « log(Ct) - ilog{rt) + log(bt), (4.13)
where rt = Rt-1 dénotes the net, nominal interest rate between t and t + 1, 7 is themoney-interest elasticity, and bt represents a serially correlated shock to money demand.
Chapter 4. Model spécification 11
4.2 The final-good-producing firm
The final good Yt is produced from an array of intermediate goods Yjt, with j €[0,1],according to the following aggregation function:
l e-i grraV djj ,6>1, (4.14)
where 9 represents the elasticity of substitution between intermediate goods.
Each final-good-producing firm maximises profits taking as given the market pricePt of the final good and the priées pjt of the input goods. It will therefore choose thequantity of intermediate goods Yjt that will maximise profits as follows:
max \ptYt- [\jtYjtdj], (4.15)
subject to the production function in (4.14). The first-order condition for Yjt impliesthe following input demand of the final-good-producing firm for intermediate good j :
9Yt. (4.16)
This input demand function will serve as the economy-wide demand for good j in theintermediate firms' sector below.
Since the final good market is perfectly compétitive, aggregate profits in the sectormust be zéro. Imposing this zéro profit condition to (4.15) and using (4.16) yields thefollowing définition of the final good's price:
4.3 The intermediate-good-producing firms
The intermediate-good-producing firm j rents Kjt units of capital, and hires Hjt unitsof labor in order to produce Yjt units of output according to the following constant-returns-to scale technology:
Yjt < AtKft (g'Hjt)1 °,a e (0,1), (4.18)
Chapter 4. Model spécification 12
where At is a stochastic (stationary) aggregate technology shock common to ail intermediate-good-producing firms, whereas g* is the non-stochastic growth rate associated to tech-nological progress.2 At is assumed to follow the autoregressive process:
log(At) = (1 - pA)log(A) + PAlog{At-{) + eAt, (4.19)
with PA G (—1,1), and tAt is a serially uncorrelated innovation to At, normally dis-tributed with zéro mean and standard déviation a A-
We solve the intermediate-good-producing firm's problem using a two step method.The first step consists in a minimisation of costs subject to the production function(4.18) where mct is the Lagrangian multiplier reflecting the firm's marginal cost ofproducing one additional unit of good:
+^r+mCt[Yjt ~AtK O HThe first order conditions for this problem are:
Kjt : %* = amcJp-, (4.21)
Hjt:^ = {l-a)mct^-. (4.22)
By replacing the first order conditions for capital and labor in the minimisation problem,we may write the total costs as:
TCt = mctYjt. (4.23)
The second step consists of maximising profits (the product of price and quantity lessthe firm's total costs). It is well known that money is neutral in a monopolisticallycompétitive framework unless some nominal frictions are added to the model. Hère,nominal rigidity is introduced following Calvo (1983). According to this spécification,firms are not allowed to re-optimise their price unless they receive a random signal.Specifically, with probability l-c/> the firm receives the signal and chooses a new price,whereas with probability (j) the firm cannot re-optimise, but may index its price to thesteady-state inflation rate, TT.3
2We could hâve written the production function as: Yjt < Kft(AtHjtY °. where At would beone single process with both trend and shock component, but for manipulation purposes, it was moreconvenient to express the idea as in the text.
3This type of indexation for firms that do not receive the signal to re-optimise follows Yun (] 996). InCEE (2005) firms that do not re-optimise index their priées to lagged inflation. Amano and Murchison(2005) combine both methods by applying partial dynamic indexation.
Chapter 4. Model spécification 13
When firm j receives the signal to re-optimise, it chooses a price pjt as well as acontingency plan for Hjt+k and Kjt+k m order to maximise its discounted, expectedtotal profits for the expected length of time during which the price pjt will remain ineffect. The profit maximisation problem is the following:
-k=Q
(Kkpjt/Pt+k - met) (4.24)
The probability that the price pjt remains in effect for t + k periods is reflected by theinclusion of 4>k.
Profit maximisation is undertaken subject to the demand for good j in (4.16). Aftersome simple algebra, the first-order condition for this optimisation problem can bewritten as:
~ G E t Y . V e k ^Pjt o i E
Because of the symmetry in the demand for their good (4.16), ail firms that receivethe signal to re-optimise choose the same price pjtj which will henceforth be denotedpt. Considering condition (4.17) for the final good's price, and the fact that at theaggregate level a fraction 1-0 of intermediate-good-producing firms re-optimise, theaggregate price index evolves according to:
P.1-0 = H^Pt-i)1-6 + (1 - <f>)p]-e. (4-26)
An interesting feature of this model is obtained from the first-order approximationof équation (4.25) and (4.26). Together, they yield a New-Keynesian Phillips curverelating the présent period's inflation rate to the expectation of future rates as well astoday's marginal costs, an indicator of the strength of économie activity:
(4-27)9
where circumflexes dénote the percentage déviation of the variables from their deter-ministic steady-state.
4.4 The monetary authority
We assume that the Bank of Canada responds to déviations of inflation ixt = Pt/Pt-i,output Yt, and money growth fxt — Mt/Mt~i by managing the nominal interest-rate Rt
Chapter 4. Model spécification 14
(Taylor, 1993). The interest-rate rule is given by:
log(Rt/R) = djog^h) + ûylog(Yt/Y) + log(fit/fi) + log(vt), (4.28)
where R, TT, Y, and \x are the steady-state values of Rt, irt, Yt, and /it rcspectively. Fur-ther, vt is a monetary policy shock that evolves according to the following autoregressiveprocess:
log(vt) = pjog(vt-i) + evt, (4.29)
where pv E [0,1] and evt is a serially uncorrelated innovation to vt with zéro mean andstandard déviation ov.
4.5 Solving the model
The model is solved using a symmetric equilibrium solution. In a symmetric equilib-rium, ail intermediate-good-producing firms make identical décisions. Let r^t = Rkt/Pt,Wt = Wt/Pt, rnt = Mt/Pt, be the real rental rate of capital, the real wage, andreal money-balances respectively. The economy is in equilibrium when the allocations(Yt,ct,Mt, Ht, Kt)^Lv priées and co-state variables (Wt^r^t, Rt,ftt, A^mc^^j^, and thedifférent shocks are such that households and both types of firms optimise, the monetarypolicy rule is satisfied, and the following market-clearing conditions are satisfied:
JoKt = / Kjtdj,
1 = / Hjtdj,J 0
li
mt =
This model admits a unique deterministic steady state in which consumption, output,investment, and real-money balances grow at the same growth rate g, while hoursworked remain constant. Since a closed formed solution to this model does not exist, themodel is loglinearized around that deterministic steady state. The model's approximatedynamics around this steady state can then be written in a state-space représentationusing Blanchard and Kahn's (1980) décomposition.4 For trended variable, let lowercasevariables correspond to their uppercase version divided by the growth rate (yt = Yt/g1),and let circumflcxes dénote the déviations of thèse variables from their steady state, sothat yt = log(yt/yss)- The model's following state-space représentation is obtained:
*See appendix E for the System décomposition
Chapter 4. Model spécification 15
st+i = $1s t + $2€t+i, (4.30)
dt = $8«t, (4-31)
where i t is a vector of state variables that includes the predetermined and exogenousvariables, dt is a vector of control variables, and et contains the random innovations.5
The éléments of the matrices $1,$2,^>3 dépend on the model's structural parameters,$.6 Since there are four sources of exogenous fluctuations in the model we will be usingfour observed time séries for estimation to prevent singularity in the détermination ofthe likelihood.
The state-space représentation can be used to estimate the underlying parameters ofthe model via Bayesian econometric techniques involving optimisation through Monte-Carlo Markov-Chain (MCMC) sampling of the posterior density function.
5 s t =s t = [ kt-i rht-i Ât bt vt zt ] ' , dt = [ yt Rt rkt ct nt wt ht \H Xt mct mt ] ' ,
= [ £At ht êvt êzt ]'6 $ H [ pA pb pz pv aA <rb cz av b y W pv py pp fi A 8 T) 6 a <j> n g }'
Chapter 5
Estimation and data
In this section, we describe the econometric approach and data used for estimating themodel's 23 structural parameters. But first, let us introduce Bayesian econometrics inits simplicity. Readers already familiar with Bayesian econometrics may skip to section5.2 without loss of continuity.
5.1 Bayesian econometrics: an overview
This section is meant for the readers not familiar with the Bayesian approach in econo-metrics. It is a basic overview of its theoretical foundations. For readers interested ina more detailed explanation, we refer you to Koop (2003) and his many références.
Baye's rule, which is central to Bayesian econometrics, is derived from the rulesof probability. Consider two random variables A and B; then the rules of probabilityimply:
p(A, B) = p(A\B)p(B) = p(B\A)p(A), (5.1)
where p(A,B) is the joint density, p(A\B) is the conditional density and p(A) is themarginal density. Rearranging the terms yields Baye's rule:
In an econometric model, we are interested in the joint probability of a parameter vector<&, and the data d, p(<&, d), which can be written as in (5.1), and then rearranged usingBaye's rule as:
Chapter 5. Estimation and data 17
The fundamental interest is with p($|d), the posterior density. Bayesians directly ad-dress the question: 'Given the data, what do we know about the parameters?' Sincep(d) does not involve $, we may rewrite Baye's rule as:
p(<fr\d) oc p(d|$)p(<ï>) (5.4)
where p($) is the prior density summarizing information the researcher has about themodel's parameters prior to estimation, and p(d\$) is the likelihood function whichis the density of the data conditional on the parameters; also referred to as the datagenerating process. Therefore, p($|d), the posterior density, is a summary of ail theinformation we hâve about the parameters. It can be seen as an updating rule wherethe data allows us to update our prior views about the parameters.
Even though p($\d) incorporâtes ail the information available about $, we are ofteninterested in certain characteristics of the distribution, for example, its mean or vari-ance. In gênerai, let g($) be a function of interest, its mathematical expectation canbe written as:
E\g($\d)] = J g($)p($\d)d$. (5.5)
An analytical calculation of the intégrais involved in (5.5) is possible only in a fewsimple cases. For more complex models, the use of a posterior Simulator is necessary.There exists différent posterior simulators. However, they are ail extensions of the lawsof large numbers or central limit theorem.
Theorem 1 : Monte Carlo Intégration
Let $ 5 for s — 1,..., S be a random sample from p(Q\d), and define
^ , (5.6)^ 8 = 1
then gs converges to i?[g(<3>|d)] as S goes to infinity.
So if it is possible to compute a random sample from the posterior, (5.6) allowsus to approximate E[g(Q\d)] by averaging the function of interest evaluated at therandom sample. However, only if S were infinité would the approximation error goto zéro. A way of reporting the approximation error is by calculating Geweke's (1992)numerical standard error. If the séquence $ 5 is simulated from independent draws froma distribution with a2 variance, the mean's (S"1 £} &S) variance is simply S~la2. In thecase where the séquence of draws is not independent, the mean's variance of a sample
Chapter 5. Estimation and data 18
where observations g($s) are correlated is S 15ff(0), where Sg(0) is the spectral densityof g(<bs) evaluated at 0.
In the next section, we explain how the Bayesian approach is applied to our model'sspécifie state-space représentation.
5.2 Bayesian estimation of a DSGE model
The model's solution was previously written in its state-space form, which will be usedto estimate the 23 structural parameters (i.e. recall équations (4.30) and (4.31)):
êt+i = $1s t + $2e«+i) (5-7)
dt = $3«t- (5-8)
The joint density we are interested in is that of the parameters in $ j , $2, and $3and the data, dt. Let $ be a vector containing ail parameters of the model, the posteriordensity of the parameters can then be written as showed in the previous section usingBaye's rule:
p($|d t)ocp(d t |*)p($) (5.9)
where j?($) is the prior density, p(rft|<3>) is the likelihood function, and p($>\dt) is theposterior density.
5.2.1 Priors
Table 5.1 gives an overview of our assumptions regarding the prior distribution of the23 estimated structural parameters. Ail variances of the shocks are assumed to bedistributed as an inverted gamma distribution (to guarantee a positive variance) asin Smets and Wouters (2003). The other priors refiect the information available fromcalibration literature, similar DSGE model estimations, or micro-founded research. Theautoregressive parameters in the technology, préférence, money-demand, and monetaryshocks, as well as the discount rate (3, capital's share in the production function a,the dépréciation of capital 5, and the Calvo parameter <fi are assumed to follow a Betadistribution which covers the range between 0 and 1. The first three hâve prior meansof 0.8 since the associated shocks are believed to be persistent, but we let the standarddéviation be 0.2 which is a bit less constraining than in Smets and Wouters (2003); the
Chapter 5. Estimation and data 19
autoregressive parameter of the monetary policy shock is believed to be less persistent soits prior mean is set at 0.5 with standard déviation O.2.1 The prior means for /3, a, and 5are set to 0.97, 0.36, and 0.025 respectively, values commonly used in the literature. Weallow /3's standard déviation to be 0.2, letting it cover a wider range, while 5's is set to0.1, and o;'s to 0.01. Notice that the standard déviation on a is much more restrictive.In previous itérations, we set it less restrictively, and the estimation led to unbelievablyhigh values for a; constraining it allows our results to stay in line with the literature.The mean for the length of price contracts (f> is set to 0.5 with standard déviation of 0.3;which allows us to fit Bils and Klenow's (2004) value (5.5 months), as well as Amiraultet al.'s (2005) who show that price fiexibility has been growing in Canada. Ail the otherparameters are assumed to follow truncated normal distributions, mostly truncated at 0,or 1 for the growth rate and the elasticity of substitution between différent intermediategoods. The monetary policy parameters' means are set close to the estimated valuesof Taylor (1993) and Ireland (2001) (pn = 1.5, py = 0.5, and pM = 0.5), but we let thestandard déviations of each cover a wider range (set at 0.5). The steady-state inflationrate is set to 1.01 following Ireland (2001), with standard déviation equal to 0.01. Themean of rj (the weight of leisure in the utility function), is set to 1.35, implying thathouseholds spend about one third of their time working, with standard déviation 0.1.The mean for the elasticity of substitution between the différent intermediate goods is6, which implies an average mark-up of price over marginal cost equal to Rotembergand Woodford's (1992) benchmark 20 per cent, with standard déviation equal to 1. Themean of x's (the capital adjustment cost parameter) is set to 15 with standard déviation5 covering the values calibrated in Ireland (2001), and Dib (2003). The mean for theparameter b, determining the steady-state ratio of real-balances to consumption, is setto 0.5 with standard déviation 0.2, matching the steady-state consumption velocityof moncy in the modcl to the average consumption velocity of M2 in the Canadiandata from 1976 to 2000 (see Dib, 2003). The mean of the interest-rate elasticity tomoney demand (7) is set to 0.05 with standard déviation 0.03; this follows Mulliganand Martin (2000) who estimate the interest-rate elasticity to money demand and showthat when interest-rates are low, 7 is also low. The growth rate g's mean is set at1.005 (representing an annual average growth rate of the economy of 2 per cent) withstandard déviation 0.002. A is a scale parameter with mean set to 3000 and standarddéviation 500.
lrThe persistence of the monetary policy shock is often estimated to be less persistent than the othershocks, see Ireland (2001).
Chapter 5. Estimation and data 20
Table 5.1: Prior distribution of parameters
Parameter
°A
06
CTz
Ov
l'A
Pb
Pz
Pv
0a5<i>V
7b
X0
.'/AP*l'y
/'/'•Il
Prior Distribution
inverted gammainverted gammainverted gammainverted gamma
BetaBetaBetaBetaBetaBetaBetaBeta
Truncated NormalTruncated NormalTruncated NormalTruncated NormalTruncated NormalTruncated NormalTruncated NormalTruncated NormalTruncated NormalTruncated NormalTruncated Normal
Mean
0.01410.01410.01410.0141
0.80.80.80.50.970.36
0.0250.51.350.050.5156
1.00530001.50.50.51.01
Standard déviation
9.0000e-0049.0000e-0049.0000e-0049.0000e-004
0.20.20.20.20.020.010.010.30.10.030.251
0.0025000.50.50.50.01
5.2.2 Likelihood
We follow Ireland (1999) in treating the capital stock as a latent variable. With latentvariables, we can exploit the recursive nature of the model and apply the KalmanFilter to evaluate the likelihood function p(dt\<&). The log likelihood for dt, knowingthat dt ~ N(dt\t-i,$Vt\t-1$'), is written:
NT 1 T
inL = —ln(2ir) - - ]T1 z t=i
1(5.10)
t=i
where dt\t-i is d*'s mean conditional on ail information available at time t — 1. For adetailed explanation of the filter, we refer the reader to the notes in Appendix F which
Chapter 5. Estimation and data 21
are based on Hamilton (1994, ch.13).
Classical econometricians will simply use the Kalman filter to construct the samplelog likelihood and then maximize it numerically with respect to the unknown param-eters of interest. However, since our sample data is relatively small, the behavior of anumerical algorithm can be very unstable. To overcome this problem, many researcherscalibrate certain parameters so as to insure convergence of the likelihood. Instead, weuse a Bayesian approach by adding prior information to make sure the behavior ofthe parameters is standard. This method is the same used by Smets and Wouters(2003). We therefore need a posterior Simulator to recover the posterior density of theparameters since analytical calculation is impossible. The next section complètes ourestimation procédure by introducing the Metropolis-Hastings algorithm within a Gibbssampler used to simulate the posterior density of the parameters.
5.2.3 The Posterior density
To simulate from the posterior density, we use the Metropolis-Hastings algorithm withina Gibbs sampler.
The Gibbs sampler is the simplest Monte Carlo Markov Chain (MCMC) technique.For each parameter $_,• in a parameter vector $, if a model is complicated, the marginaldistributionp($j\d) is not standard. However, the conditional distribution p(Qjl^j-i, d)may be, where &j-i represents ail the model's parameters excluding <&j. If we cansimulate a candidate from the conditional p($j|$j-i,d), we can generate a randomséquence with the following:
} one itération
*3
one round (5-11)
The Markov chain described above converges to its stable distribution p(&\d).2 If theconditional distribution p($j|$j_i,d) is not standard (as in our case), we may use theMetropolis-Hastings algorithm to recover it.
The Metropolis-Hastings algorithm developed by Metropolis and al. (1953) andHastings (1970) can be used to simulate draws in non-standard distributions. It is a
2See Geman and Gernan (1984), and Gordon and Bélanger (1996), for a more detailed explanation.
Chapter 5. Estimation and data 22
Monte Carlo simulation-based technique which uses Markov Chain properties to assureconvergence of the simulated density; thus it is referred to as a MCMC technique. Formore intuition as to its theoretical foundations, we refer the interested reader to Chiband Greenberg (1994).
To simulate a new draw in the chain, we use a known density, henceforth referred toas the candidate generating density (CGD). Then, a probabilistic rule is used to décidewhether or not the new candidate is accepted.
Let the CGD be written as ç($j,$^). If the last value of the chain is $j , thecandidate ^ is simulated from the distribution of the CGD. The latter candidate isthen accepted with probability a (<&.,•, <£>'•), where:
' " \(*l4)«(*»}) JOnce the new candidate $'• is simulated, (5.12) is used to calculate the probabilitythat it will be accepted. A simple way of applying this rule is to simulate a réalisationU from a uniform distribution (0,1); the candidate is accepted if U < a. Thus, wemay summarize the algorithm to simulate réalisation <£* of the Markov chain with thefollowing steps:
1. Given the previous value <&*rl, simulate a candidate <î from the CDG
2. Calculate a ^ , ^ - )
3. Simulate U from a Uniform distribution (0,1). If U < a, fix $* to $£; or clse, fix
*} to S}"1.
4. Repeat
It can be proved (Chib and Greenberg, 1994) that the Markov chain described bythis algorithm is ergodic and converges to p(<f>j\dt)- Then, it is just a inatter of simplearithmetics to calculate point estimate of the mean or variance of the parameters usingTheorem 1 in section 5.1.
A last explanation is needed hère as to the choice of a CDG. We use the one suggestedby Metropolis and al. (1953) where <?($_,•, $^) = /($^- - $,), where /(.) is a knowndensity. In this case, the candidate is simulated from a random walk:
Chapter 5. Estimation and data 23
where z ~ f(z)- With the random walk CDG, candidate draws are taken in randomdirections from the current point. If the innovations of the random walk come froma symmetric distribution, note that f(z) = f(—z), then q($j,3^-) = f($j — $!•) =ç($j-,$j). If that is the case, the expression q($'j,$j)/q($j, $^) in (5.12) disappearsand the probability of accepting the new candidate is simply:
(5.14)
In other words, if the targeted density p(<&j\dt) is higher with $ - than it is with $ J ; theprevious iteration's value, then we accept the candidate. However, if the value of thedensity lessens of say ô, then it is only accepted with probability (1 — 8).
The form of the CDG is determined by the choice of density for z. The multivariateNormal is the most common and convenient choice. The mean of the Normal is deter-mined by $j in 5.13 and the researcher must choose the covariance matrix to insurethat the acceptance probability is neither too high nor too low. To find the covariancematrix, one can simply experiment with différent values until a reasonable acceptanceprobability is obtained.3
5.3 Data
We estimate the model's parameters using a subset of the control variables dt in (5.8)for which data is available. Since the model is driven by four stochastic processes,we estimate it using data for four séries to prevent singularity in the détermination ofthe likelihood. The data are taken from CANSIM, Statistics Canada's database. Theséries include output, real money balances, a short-term interest-rate, and inflation.Output is measured by real, final domestic demand. Real money balances are measuredby dividing M2 (currency and ail chequable notices and personal term deposits) bythe GDP deflator. The short-term interest-rate is the three-month Treasury Bill rate.Finally, inflation is the gross rate of increase in the GDP deflator. Further, output andreal money balances are expressed in per capita terms using the population aged 16 to64.4 Ail data are logged and rendered quarterly (by taking a three month average whennecessary) before estimation.
3There is no gênerai rule as to the idéal acceptance probability. However, if it is roughly 0.5 one isunlikely to go wrong. (Koop, 2003, 97-98)
4The séries used are V1992078 for output, V37128 for M2, V122531 for the interest-rate, V1997756for the GDP deflator, and V2091037 for population which can be found at www.statcan.ca
Chapter 6
Estimation results
We proceeded to several preliminary replication rounds in order to fînd the convenientcovariance matrix for the random walk CDG. Once convergence was obtained we thenproceeded to calculate the posterior mean for each parameter under the Monte CarloIntégration theorem 5.6. This procédure was followed for both of our sample periods."Fable 6.1 présents Bayesian posterior means of the model's 23 parameters estimated onour two sample periods (entire sample: 1981Q1-2000Q4, subsample: 1991Q1-2000Q4).Geweke's (1992) numerical standard crror (NSE) was also calculated and in the entiresample (subsample), the NSE associated to 12 (15) parameter estimâtes is under 1 percent of the absolute value of the parameter's posterior mean, associated to 20 (22) isunder 5 per cent, and associated to 23 is under 17 per cent (11 per cent).1
Focusing first on the parameters of the policy rule, the estimâtes of TT, 1.0182 forthe entire sample and 1.0107 for the subsample, translate into annualized, steady-stateinflation rates of 7.5 per cent and 4.3 per cent respectively. Further, the interest rateresponse to inflation, mcasured by pn, is much larger at 1.3926, when estimated withour subsample, versus 0.7130 when estimated with the entire sample. Thèse estimâtesirnply that the Bank of Canada has been more aggressive in fighting inflation duringour subsample period, which accords well with our priors and the introduction of theinflation targeting policy in 1991. Ireland (2001), and Clarida, Gali, and Gertler (1998)find similar results for the before-and-after Volkner period at the Fédéral Reserve in theUnited States. Further, the interest rate response to variations in moncy growth, pM, isalso slightly more important in our subsample, another indication of the increased ag-gressiveness of the Bank to inflationary pressures. On the other hand, in both cases, theinterest rate response to variations in output, py, is practically zéro, a resuit consistent
'in appendix H is a table with both posterior mean estimâtes as well as NSE values.
Chapter 6. Estimation results 25
with the literature (see Ireland, 2001, Smets and Wouters, 2003, and Dib, Gammoudi,and Moran, 2005).
Interestingly, ail the other parameters seem to remain stable over time, at valuesressembling those commonly calibrated or estimated in DSGE literature. The discountrate (3 is estimated in the entire sample (subsample) at 0.9877 (0.9884), a value closeto the ones used in the calibration literature. The dépréciation of capital 5 is slightlyhigher than what is normally used in calibration (0.025) at 0.0335 (0.0330). The weighton leisure in the représentative household's utility function 77 is 1.3105 (1.3376) whichimplies that the household spends about one third of its time working; again a commonlycalibrated value. Capital's share in the intermediate good's production is close to 0.36,another commonly used value. The elasticity of substitution between the différentintermediate goods 9 is 6.8217 (6.5425), which implies an average mark-up of price overmarginal cost equal to Rotemberg and Woodford's (1992) benchmark 20 per cent. Theestimate for b, the parameter determining the steady-state ratio of real balances toconsumption is 0.5205 (0.5320); while that of 7, the elasticity of substitution betweenconsumption and real-money balances is 0.0613 (0.0701), similar to the estimated valuesof Dib, Gammoudi, and Moran (2005) for the Canadian economy. The estimâtes of(f), 0.7005 (0.6792), imply that firms, on average, re-optimise their priées about onceevery 9 months. The préférence, money-demand, and technological shocks are ail verypersistent, whereas the monetary policy shock is much less persistent. Again, this isconsistent with findings in Dib, Gamoudi, and Moran (2005).
The fact that therc is no significant change to key parameters, apart from thechanges in the monetary policy rule, seems to indicate that the model is stable overtime. The variations of the estimated parameters in the policy rule simply capture thefact that the monetary policy has changed in 1991. The model seems stable, but howis it to be formally judged? Chapter 7 discusses how we answer this question.
Chapter 6. Estimation results 26
Table 6.1: Posterior Means of Parameters
Parameter Estimated with the entire sample Estimated with the subsample
(3 0.9877 0.9884
p A 0.9999 0.9999
aA 0.0150 0.0134
pb 0.9968 0.9747
ab 0.0131 0.0131
Pz 0.8560 0.7658
az 0.0181 0.0150
pv 0.2883 0.3732
av 0.0125 0.0141
a 0.3630 0.3604
5 0.0335 0.0330
0 0.7008 0.6792
ri 1.3105 1.3376
7 0.0613 0.0701
6 0.5205 0.5320
X 21.556 19.616
0 6.8217 6.5425
g 1.0048 1.0047
A 3131.1 3023.1
p^ 0.7130 1.3926
py 0.0015 0.0026
p M 1.2057 1.5289
7T 1.0182 1.0107
Chapter 7
Forecasting comparison
As we mentioned in the previous section, most parameters hâve remained stable throughthe monetary policy shift of 1991, although the parameters in the interest rate ruleseem to hâve changed. One way of establishing if the model is stable over time is bycomparing its out-of-sample forecasts when estimated over the two différent sampleperiods. This also permits us to evaluate the importance of the variation in parameterson the forecasting ability of the model. One-step-ahead forecasts are evaluated usingthe prédictive density:
p(dt+l\dt) = J p(dt+1\dt,$)p($\dt)d$, (7.1)
with the optimal mean satisfying:
E[dt+i\dt] = jp(dt+1\dt)ddt+u (7.2)
which can be calculated through our state-space représentation by averaging over theforecast of each draw <3>\ So, for each of the replications of $ kept from our MCMCposterior Simulator, calculate dt+l from the observation équation of the state-spacereprésentation:
dt+1 = P($)st, (7.3)
then, average over ail dt+1 to hâve a one-step-ahead forecast.
Figures 7.1 to 7.4 show for both samples the one-step-ahead forecasts associatedwith the four time séries used to estimate the model. In each graph, there are twopanels, the upper panel gives a view of the entire period (1981Q1-2005Q1), while thelower panel enhances the latter part of the upper panel. Note that the data used forestimation runs from 1981Q1 to 2000Q4 (entire sample) and from 1991Q1 to 2000Q4(subsample), so we are comparing one-step-ahead forecasts with the actual available
Chapter 7. Forecasting comparison 28
data starting with 2001Q1 up to 2005Q1. Also, note that for each new observationadded, the model is re-estimated.
10.85
10.8
10.75
10.7
10.65
10.6
10.55
10.5
10.45
Figure 7.1: Forecasting Outputlog GDP
— Subsample torecasts— Enlire sample forecasts— Actual data
1 !
1985 1990 1995 2000 2005 2010
10.6B
Subsample forecasts- Entire sample lorecasts
-— Actual data
1998 1999 2000 2001 2002 2003 2004 2005 2006
Chapter 7. Forecasting comparison 2!)
10.2
10.1 -
10 -
9.9
9.8 -
9.7 -
9.61980
Figure 7.2: Forecasting Real Money Balanceslog real money balances
i i i
i
Subsample forecasts- Entire sample forecasts
—— Aclual data
i I
1985 1990 1995 2000 2005 2010
10.14 -
10.12 -
10.1
10.08
10.06 -
10.04 -
10.02
! 1 1 1
/ - ^i i i
i
/ ^ /
Subsample forecasts- Entire sample forecasts
— Actual data
i i101998 1999 2000 2001 2002 2003 2004 2005 2006
Chapter 7. Forecasting comparison 30
Figure 7.3: Forecasting the Interest RateAnnualized interest rate
20
15
II)
- Subsample forecasts— Entire sample (orecasts—•— Actual data
1980 1985 1990 1995 2000 2005 2010
6.5 -
6 -
5.5 -
5 -
4.5-
4 -
3.5
3
2.5
~ Subsample forecasts- Entire sample forecasts
— Actual data
1.5'—1998 1999 2000 2001 2002 2003 2004 2005 2006
Chapter 7. Forecastmg companson :M
14
12
10
Figure 7.4: Forecasting the Inflation RateAnnualized inflation rate
0
-2
4
-6 '—1980
Subsample forecastsEntire sample forecastsActual data
1985 1990 1995 2000 2005 2010
A
16998
— Subsample forecasts- Entire sample forecasts
— Actual data
1999 2000 2001 2002 2003 2004 2005 2006
Chapter 7. Forecasting comparison 32
Simply by glancing at the graphs, we can see that both sample period estimâtesrender similar forecasts.1
A more robust way of comparing the forecasting ability of the model under the twodifférent samples is by using Bayesian forecasting model comparison. Let the modelestimated on the entire sample period be Model A, and let Model B be the one estimatedon the subsample, and let p(A\dt) and p{B\dt) be the probabilities associated to eachmodel before the forecasting exercise. Conditional on dt, it is possible to dérive theprédictive density p(dÉ+11A, dt) a,ndp(dt+i\B,dt). Once dt+i is observée!, the informationset becomes Dt+i = (dt+i,dt), and the probability of model A conditional on dt+i is:
p(A\dt+1) = —. Pid^\Ajt)p[A\dt)_ _p(dt+l\Ajt)p(A\dt)+p(dt+l\Bjt)p(B\dt)
where p(A\dt) and p(B\dt) are our priors on each model. We did the exercise with 0.5priors on each model. Then, for the next period, we hâve:
P(A\it+2) =p(dl+2\A,dt+i)p(A\dt+l) + p(dt+2\B,dt+1)p(B\dt+1)
so our first posterior probability p(A\dt+i) becomes the prior for our next, and so on.
Table 7.1 présents model probabilities for Model A and Model B on our entire fore-cast horizon. Notice that the probability associated to the last period is much higher inthe subsample so it indicates that it would be the preferred model. However, depend-ing on the date of the last observation added, we might prefer the model estimated onthe entire sample period (see the probability associated with period 2002Q4 or 2003Q1for example). Neither model outperforms the other definitively, which supports theconclusion that the model is in fact stable over time.
1In appendix I, you will find tables presenting the forecasting values for each period using both theentire sample and subsample estimâtes with the associated standard déviation in brackets.
Chapter 7. Forecasting comparison 33
Table 7.1: Probability of the model's forecast
Quarter forecasted Model A (Entire sample) Model B (Subsample)
0.56610.71890.78230.36340.42040.25590.43230.35600.29140.70030.60350.73410.56010.44550.57610.70370.7244
2001Q12001Q22001Q32001Q42002Q12002Q22002Q32002Q42003Q12003Q22003Q32003Q42004Q12004Q22004Q32004Q42005Q1
0.43390.28110.21770.63660.57960.74410.56770.64400.70860.29970.39650.26590.43990.55450.42390.29630.2756
Chapter 8
Conclusion
This paper has developed and estimated a closed economy DSGE model of the Canadianoconomy using a Bayesian approach. We provide estimâtes of its structural parametersover two subsamples separated by thc introduction of inflation targeting in 1991 bythe Bank of Canada, in order to establish whether the model is stable over time. Ourestimâtes suggest that the model is able to capture the monetary policy shift. In fact,the estimated parameters of the Taylor rule followed by the monetary policy appearto change significantly in 1991 suggesting a more aggressive reaction from the Bank toinflationary pressures. By contrast, the other parameter estimâtes are unchanged fromone sample to the other. We also compare the one-step-ahead forecasting ability of themodel estimated over the two subsamples and find that neither outperforms the otherwhen compared to the actual data. Finally, to provide a more complète analysis, weundergo a Bayesian model comparison test based on the model's prédictive densitiesand find that the preferred model dépends on the date of the last observation added.Overall, our results support the view that the estimated DSGE model is stable, whichis very promising for further research in developping and estimating DSGE models forforecasting and policy analysis purposes. Considering that the closed economy hypoth-esis in the context of the Canadian economy might not be acurate, a next challenge forresearchers would be to adapt the model to an open-economy to provide a complètestability analysis.
Bibliography
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Appendix A
Symmetric equilibrium équations
Defîne : rkt = Rkt/Pt
Market clearing conditions : Kt = J^ Kjtjt, Ht = $ Hjtjt, Yt = JQ Yjtjt
y— I
7
= A t -A,
'/1-Ht
Kt
= (1 - a)m C i|
Yt =
(A.l)
(A.2)
(A
Kt+2 *
(A
/ A1 A
(A
(A
3)
rïl' J .4)
6)
7)
(A.8)
(A 9)
Appendix A. Symmetric equilibrium équations 40
log{Rt/R) = tijog{nt/ir) + ^/o5(F t/y) + ^logijH/fi) + log{vt), (A.13)
Zo^(^) = pzlog(zt-i) + ezt (A.14)
(1 - pb)log(b) + pblog{bt-i) + e« (A.15)
= pjog(vt-i) + evU (A.17)
Appendix B
TYansformed (stationary) System
Since certain variables hâve an upward time trend, we define them in terms of station-ary (detrended) variables :
et = Ct/g* h = Kt/g*
it = W Vt = Yt/g*= mt ht = Ht
= wt Xt = V
So as to write:
^ZtC\ ^ = A, (B.l)ct
7 + b?mt'1
>f mt
(B.2)
,,iL + 1 - 5 + ( ^ - ^ j ^ i i a - l ( ^ i ± i _ p j
(B-4)
(B.5)
2N
rkt = amct-t (B.6)«t
Appendix B. Transformed (stationary) System 42
wt = (1 — a)mct— (B-7)
Jhit'] (B.8)0 - 1 £*$:£.„.
yt = Atk?{ht)l-a (B.9)
^A;<+i = (1 — ô)kt + yt — ct — — (—-—- — g)2kt (B.10)
•e + 4>T< (B.ll)
log(Rt/R) = iïjog{irt/n) + dylog(yt/y) + log(fxt/fi) + log(vt), (B.13)
log(zt) = pjog(zt-i) + ezt (B.14)
ôt) - (1 - pb)log{b) + phlog(bt-x) + ew (
^ ) ^ ( ^ l ) + pAlog{At-i) + e^t (
= pjog{vt_i) + evt, (
Appendix C
Steady-state solutions
The following steady-state solutions are derived from équations B1-B17.
y. = grc (Cl)
R=l (C2)
(C3)
(C.4)
(C.5)
(C6)
(C.7)
(C8)
(C9)
(CIO)
(Cil)
We now know h/(l — h) = x, so h = s/(a; + 1), we know x so we know h, from whichwe may find ... c = c/y * y, k = k/y * y, A = Ac/c, m = Xm/X, w = (1 — «)mc *
7
c
2/
Am :
h1 -
y =
iy =
me
• * =
A;
2/
= Ac
2/
= ( ! •
ô - l
- - 1 + ^
amc
fcy
i
•7T7( - /3)^t
Ac(l — a)mcr}(c/y)
Yy-«A^h
- a)mc(y/h)
Appendix D
Linéarisation : Blanchard and Kahn
Static équations:Aft = Bdt + Cxt (D.l)
Dynamic équations:Ddt+1 + Eft+i = Fck + G ft + Hxt (D.2)
with A, B, C, D, F, G, H being matrices, f is a vector of flow variables, d is a vector ofdynamic variables and x is a vector of shocksStatic Equations
ijt + (a - l)ht = akt + Ât (D.3)
fxt - jtt = rht - rht-i (D.4)
Vt ~ rkt = k~ rnct (D.5)
- [1 + Ac(7 - l)]ct = 7Ât + Am(^^) (7 - ï)rht + \m{—-)bt - -yzt (D.6)7T 7T
R 'V — 1 - 7T — R 'M — 1 1 7T — /9 1 1 - v
& A c 2 c A + ( A ? r i ^ ^ ^ + )m + ( A ? n ^ ) 6 t - ^ (D.7)7T — p 7 7 T 7 7 7 T 7 7
——ht -wt = \t (D.8)l - h • '
yt — wt — ht = —met (D.9)
Dynamic Equations
(D.12)
Appendix D. Linéarisation : Blanchard and Kahn 45
kt+lgk -(l-ô)kkt + yty - ctc
Af+i ~ iït+\ = A4 — Rt
iht = rht
(D.13)
(D.14)
(D.15)
The previous équations may be written in their matrix form as in D.l and D.2 withthe following éléments in each lines/columns:
Vt
(59) /(60)(61)
A - (62>" (63)
(64)(65)
(
(
(
(
:
)
i
)
)
)
i
(66) \-û
(59)(60)
(61)
B - <62>(63)(64)(65)(66)
(59)(60)(61)
c _ (62)(63)(64)(65)(66)
(
\
\
y
k(X
0J0
0000
Ât
10000000
Rt
00000
JT-/9
001
îht-i
0- 1000000
A'Xm-
ht00
- i
0
000(J
Ât
000
71
1
00
k000
^ 7 •
000
- 1
râct
00
- 1000
- 10
_ i7
—
;
Vt
00000001
Ct
000
( 7 -^ ( ^
00
0
À77l(')
A m ^7
000
- 7^
f)
00
l)Ac
• )
fnt
01
0
- 1 )- 7T-/3
•ïï
000
\
/
Kt
0- 1
00000
\
7T
+ i
^ t
00000- 1- 10
ht
a- 10000h
l-h
- 1
0
h0i
00000
-d
Appendix D. Linéarisation : Blanchard and Kahn 46
D
E =
G =
(68)
(69)
(70)
(71)
(72)
(68)
(69)
(70)
(71)
(72)
(68)
(69)
(70)
(71)
(72)
(68)
(69)
(70)
(71)
(72)
/
V
—
m+i Rt-
( °0<pyg
k
0
0
l ok
( o-<P9
(1-5)*0
m Rt/ 0 0
0 0y o0 - 1
V 0 0
00000
kt+0
s)-gk00
t-i
-(1
ht+
0f3rk
000
rht-i
00000
ht0000
0
Q
00
—c
00
+
1
Xt
0101
0
* t
10000
?))
0_â
000
1
( 1 -
wt
00
00
0
mt
00 /
001
/300
- 1
0
mct
f3<t>)(l-
0000
ht
00000
3(r
i
0)
A<000
0
0
. t+1
0
fc + 1
0i
0
00000
rht
o N0
0
0
1 )
\
/
-5)
îk+i0
0000
rnct+1
0
0000
Am0 ^
0
0
0
o J
mt+i0 >
000
o )
Appendix E
Blanchard and Kahn'sdécomposition
Static équation:(E.l)
Dynamic équation:Ddt+1 + Eft+1 = Fdt + Gft + Hxt (E.2)
Shocks:xt+1 = pxt + Çt+i,Ç = N{Q,(T) (E.3)
First step : isolate f in the static équation
ft = A~lBdt + A~lCxt (E.4)
One period ahead:ft+l = A-xBdt+l + A~lCxt+1 (E.5)
Replace ft and ft+i in dynamic équation:
Ddt+1 + E[A-lBdt+l + A~lCxt+1] = Fdt + G[A~lBdt + A~lCxt] (E.6)
Remember : Et(xt+\) = pa;*, that we may replace in previous équation:
Ddt+i + ^[A-^dt+i + A-^/twt] = Fdt + GfA-^dt + A~lCxt} (E.7)
Put same variables together:
[D + ^M-^BH+i = [F + GA-lB]dt + [GA'lC - EA-lCP}xt (E.8)
Appendix E. Blanchard and Kahn's décomposition 48
Isolate dt+i
dt+i = [D + EA~lB]-l[F + GA-lB}dt + [D + EA-lB)-l[GA-lC - EA~xCp\xt (E.9)
Compactly:
+ Lxt (E.10)
Diagonalize K: K = M~lNM, with M being eigenvectors and N being eigenvalues, N
I with JVi < 1 and JV2 > 1 being vectors.0 N2 /
M = [ I with Ma being vectors. Also write L= I\M2i M22)
3 B \L2
So dt+i = M-^NMdk + Lxt
Premutiply by M : Mdt+l = NMdt + MLxt or
/ M n M12\ /iVx 0 \ / M u M12\ / M n M12\ / L A\,M21 M22)'
t+l~ \0 N2) \M21 M22) t + \M21 M22) \L2) *
Define : Qx = MnLi + Mi2L2 and Q2 = M2XLX + M22L2
Also define dt = [kt At]', where fct are predetermined variables and At are not predeter-mined. Therefore, associate Ni to kt and N2 to Àf.
Define : kt = MuA;t + M12A( and Ât = M2ikt + M22Xt
So: fct+i = ^ 4
Xt+1 = N2\t + Q2xt
We must cancel out the effect of unstable eigenvalues A2 > 1, so /zmw^ooAf = 0
Knowing that At+i = A A* + Q2^«) w e niay isolate Xt :
Ât = N^Xt+i - N2lQ2xt (E.ll)
Recursively:Ât = N2
l(N2l\t+2 - N^Q^t+i) - N2
lQ2xt (
By continuous recursion:
.s-0
Appendix E. Blanchard and Kahn 's décomposition 49
Remember that : lim^^tOON2^n+ Xt+n+i = 0, and Et(xt+B) = paxt
Or more compactly:
Xt = uxt (E.15)
Let's go back to the équation: Xt = M2ikt + M22Xt Isolate At
Xt = M22l Xt - M22
l M2l kt (E. 16)
Replace Xt:Xt = M22uxt - M22M2ikt (E.17)
Write more compactly:
We may now replace At:
Mufci+i + M 1 2 A m = NxMlxkt + QlXt
Replace Ai+1:Mukt+i + M12[s2xt+1 + aifee+i] = iViMu/ct + Q ^ (E.20)
Isolate &t+i:
fct+1 = [Mn + MuaiJ^^Mufc t + [Mu + M ^ s i ] " 1 ^ ! - Ml2s2p]xt
Compactly:
fct+1 = s3kt + SiXt (E.22)
We may go back to the static équation and replace A:
ft = A'^Bdt + A-xCxt (E.23)
(E.24)
+A-lCxt (E.25)
ft = A~1B (^kt + lA-'C + A-'B ( s°Vt (E.26)
More compactly:ft = s5kt + s6xt (E.27)
Appendix E. Blanchard and Kahn 's décomposition 50
Matrix form of state équation:
UJ + [lMatrix form of observation équation:
(ft\ fs5 sA fkt\\Xt) \sr s2) \xt)
State-space form:
Observation équation:dt = $3St (E.28)
State équation:st+l = $ l S t + $2ef+1 (E.29)
Appendix F
Using the Kalman filter forlikelihood évaluation
State-space représentation of the model:
st+i = $ist + $2ei+i (F.l)
dt = $ 3 * (F.2)
where et ~ JV(O, £).
The Kalman filter is used to calculate linear least square forecasts of the state vector
on the basis of data observed at time t. It is used to generate projections recursively
from si|0, s2|i, to
Associated to each projection is the mean square error (MSE):
Pt+i\t = E\{st+l - «t+i|t)(st+i - h+x\t)'] (F.3)
where st+i\t — i?[si+i|d(] is a linear projection of st on dt and a constant.
To start the recursion, we start with ,§i|o which is a forecast of si based on no
observation, and is simply the unconditional mean of sy.
ii|o = E(Sl) (F.4)
Pll0 = E[(Sl - E(Sl))(Sl - E(Sl))'} (F.5)
From the state équation, the unconditional mean is:
E(Sl) = ^($is*_i) (F.6)
Appendix F. Using the Kalman filter for likelihood évaluation 52
And the unconditional variance is:
E(sts't) = £[(<Mt-i + $2e,)(sU$i + 2 ) ] (F-7)
Letting V dénote the covariance matrix of s,
V = $ 1 ^ + $2S$2 (F.8)
From which we can recover V,
vec(V) = [Ir2 - $! ® «Êx]"1 * wec($2S$'2) (F.9)
For a projection of dt, from the observation équation we hâve,
dt\t-i = <Mt|*-i (F.10)
And the forecast error is:
dt - dt\t-l = *3St - $3%- l = *3(5* - St|4_i) (F.H)
And associated MSE:
E[(dt - dtlt^)(dt - dti
Cross-products disapear and we are left with
E[(dt - 4 4
Next, updating the linear projection:
st\t = «t|t-i + [£?(«< - st|*-i)(dt - ^| t-i)'][S(dt - dt\t-i)(dt - dt\t-i)'} l * (dt - d\\t-i)
= st\t-i + ^ | t_i^($ 3Vt| t- i*3) * (d/ - 4t|i-i)
Then, forecasting:
*iâ*|t_i + $!Ki t - i^C^Vi i t^ )" 1 * (dt - dt|t_i) (F.14)
The log likelihood for dt, knowing that dt ~ iV(rfÉ|t_l5 4»3Vi|t—1 *3) is written:
NT "\ r \ T
inL = -—ln(2ir) - - ^ ^ 1 * 3 ^ 1 - -1 l t=i ^ t=i
Appendix G
Posterior means and NSE
Table G.l: Posterior means' point estimate
Parameter
0PA
o~A
Pb
o'b
Pz
°zPv
Ov
aS
<(>
V
7b
X0
9AP-n
PvPi''IX
Entire sample posterior mean (NSE)
0.9877 (0.0012175)0.99989 (4.4849e-013)0.015029 (0.00012892)0.99676 (0.00028062)0.013093 (3.536e-005)0.85604 (0.0058517)
0.018068 (0.00022823)0.28827 (0.013622)
0.012468 (0.00010151)0.36303 (0.0009598)
0.033345 (0.00079644)0.70047 (0.0066685)
1.3105 (0.005701)0.0613 (0.0021902)0.52049 (0.0060713)
21.556 (0.29805)6.8217 (0.094388)
1.0048 (0.00010255)3131.1 (18.18)
0.71299 (0.063402)0.0014543 (0.0002385)
1.2057 (0.019413)1.0182 (0.0011923)
Subsample posterior mean (NSE)
0.98841 (8.9185e-005)0.99998 (7.3948e-005)0.013418 (0.00035304)0.97471 (0.00085411)
0.013063 (4.8457e-005)0.76577 (0.0028746)
0.014977 (6.9604e-005)0.37322 (0.0037299)
0.014096 (5.2664e-005)0.36038 (0.00053023)
0.033034 (0.00039341)0.67916 (0.0033531)1.3376 (0.0037022)
0.070124 (0.0017171)0.53195 (0.0042428)
19.616 (0.13101)6.5425 (0.051576)
1.0047 (6.1826e-005)3023.1 (18.714)
1.3926 (0.020655)0.0025882 (0.00027827)
1.5289 (0.017887)1.0107 (0.00081707)
Appendix H
Forecasting results
Table H.l: Forecasting results using the entire sample period parameter's estimâtes
Period Output Inflation Interest rate Real Money balances
2001Q12001Q22001Q32001Q42002Q12002Q22002Q32002Q42003Q12003Q22003Q32003Q42004Q12004Q22004Q32004Q42005Q1
10.752(0.0021)10.761(0.0023)10.767(0.0025)10.774(0.0024)10.774(0.0035)10.768(0.0035)10.769(0.0034)10.783(0.0030)10.782(0.0031)10.785(0.0033)10.807(0.0024)10.808(0.0030)10.814(0.0022)10.821(0.0035)10.822(0.0038)10.83(0.0030)10.839(0.0027)
2.6255(0.3583)2.2589(0.3406)0.5612(0.4038)-2.4570(0.5362)-3.1228(0.5833)1.6501(0.3785)4.3905(0.5447)1.3670(0.3941)2.9930(0.4340)3.8957(0.5490)-0.5987(0.4876)2.2182(0.3979)0.9635(0.3294)2.2182(0.4015)2.8295(0.4816)1.8933(0.3658)1.1651(0.3644)
6.1782(0.3093)5.4271(0.3336)4.9700(0.4073)4.3492(0.4011)2.5032(0.4477)2.8295(0.4246)3.4026(0.3627)3.6078(0.3833)3.4026(0.3725)3.6900(0.3837)3.8133(0.3453)3.4026(0.3639)3.4846(0.3125)2.8295(0.3441)2.5439(0.4154)2.9930(0.4083)3.2386(0.3802)
10.03 (0.0016)10.035 (0.0019)10.047(0.0018)10.065(0.0018)10.102(0.0032)10.095(0.0032)10.077(0.0033)10.096(0.0027)10.090(0.0029)10.081(0.0030)10.116(0.0020)10.113(0.0029)10.112(0.0019)10.114(0.0034)10.12(0.0038)10.124(0.0029)10.128(0.0023)
* Note that the interest rate and the inflation rate are expressed in annualizedvalues, whereas output and real-money balances are logged.
Appendix H. Forecasting results 55
Table H.2: Forecasting results using the subsample period parameter's estimâtes
Period
2001Q12001Q22001Q32001Q42002Q12002Q22002Q32002Q42003Q12003Q22003Q32003Q42004Q12004Q22004Q32004Q4
2005Q1
Output
10.761(0.0024)10.763(0.0028)10.763(0.0023)10.762(0.0040)10.769(0.0027)10.777(0.0035)10.781(0.0038)10.786(0.0027)10.791(0.0036)10.799(0.0035)10.809(0.0020)10.813(0.0026)10.823(0.0024)10.826(0.0026)10.831(0.0039)10.839(0.0031)10.85(0.0024)
Inflation
1.9745(0.3641)1.6501(0.4056)0.0800(0.3647)-2.2763(1.455)-2.7707(0.5369)1.651(0.5038)
3.4436(0.6871)0.8829(0.4470)2.2996(0.6813)2.9930(0.6243)-0.8254(0.4111)1.5691(0.4742)0.5210(0.4144)1.5691(0.4745)2.1776(0.7353)1.4074(0.5898)0.7622(0.3865)
Fnterest rate
5.5103(0.1880)4.8456(0.1958)4.5145(0.1849)4.0604(0.1948)2.6255(0.2705)2.7479(0.2318)3.1567(0.1922)3.3206(0.1872)3.1976(0.1909)3.3616(0.1647)3.4846(0.1706)3.1158(0.1682)3.1567(0.1654)2.6255(0.1761)2.3810(0.1903)2.6663(0.1694)2.9521(0.1558)
Real Money balances
10.029 (0.0017)10.034 (0.0019)10.046(0.0017)10.064(0.0030)10.104(0.0021)10.096(0.0024)10.078(0.0029)10.096(0.0020)10.091(0.0029)10.083(0.0026)10.115(0.0015)10.113(0.0021)10.112(0.0020)10.116(0.0020)10.123(0.0031)10.124(0.0026)10.128(0.0018)
