Business cycle asymmetry: state-space models with Markov switching.

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Geoﬀrey Coke

B.Sc., University of Victoria, 2004 M.Sc., University of Victoria, 2009

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF ARTS

in the Department of Economics

c

� Geoﬀrey Coke, 2012 University of Victoria

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Business Cycle Asymmetry: State-Space Models with Markov Switching by Geoﬀrey Coke B.Sc., University of Victoria, 2004 M.Sc., University of Victoria, 2009 Supervisory Committee

Dr. Graham Voss, Supervisor (Department of Economics)

Dr. Ken Stewart, Departmental Member (Department of Economics)

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Supervisory Committee

Dr. Graham Voss, Supervisor (Department of Economics)

Dr. Ken Stewart, Departmental Member (Department of Economics)

ABSTRACT

This thesis extends Kim and Nelson’s (1999) plucking model for real GDP to include correlated innovations. The resulting correlated innovations unobserved com-ponents (UC) model allows for both asymmetric transitory movements and correlation between the permanent and transitory innovations. Applying the extended model to U.S., Canadian and Australian GDP, I show that the GDP series can be usefully decomposed into a permanent component, a symmetric transitory component, and an additional occasional asymmetric transitory shock. Incorporating correlated in-novations in the model changes the allocation of volatility between the permanent and transitory components. For the U.S., correlated innovations were found to be significant and the permanent component accounted for a larger share of the volatil-ity in GDP. For Canada and Australia, correlated innovations were not significant and the fitted model produced smooth permanent component and volatile transitory component estimates.

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List of Tables

Table 3.1 U.S. Data: Maximum Likelihood Estimation of the Models (Model 2 and 3 ML estimates of σu1 fell on the boundary (σu1=0). To

calculate standard errors σu1 was set equal to 0.) . . . 15

Table 3.2 Canadian Data: Maximum Likelihood Estimation of the Models (Model 1 and 2 ML estimates of σu1fell on the boundary (σu1=0).

To calculate standard errors σu1 was set equal to 0.) . . . 16

Table 3.3 Australian Data: Maximum Likelihood Estimation of the Models (Model 3 ML estimate of σu1 fell on the boundary (σu1=0). To

calculate standard errors σu1 was set equal to 0.) . . . 17

Table 3.4 Maximum Likelihood Estimation of the Models for Time Periods 1947:1-1983:4 and 1984:1-2011:1 . . . 26

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List of Figures

Figure 3.1 U.S. Data: Estimated Trend and Transitory Components and Probabilities of Negative Shocks for the Correlated Plucking Model 21 (a) Correlated Plucking Model: Real GDP and Its Trend Component 21 (b) Correlated Plucking Model: Transitory Component . . . 21 (c) Correlated Plucking Model: Probabilities of Negative Shocks . . 21 Figure 3.2 U.S. Data: Estimated Trend and Transitory Components and

Probabilities of Negative Shocks for Kim and Nelson’s Plucking Model . . . 22 (a) KN Plucking Model: Real GDP and Its Trend Component . . . 22 (b) KN Plucking Model: Transitory Component . . . 22 (c) KN Plucking Model: Probabilities of Negative Shocks . . . 22 Figure 3.3 Canadian Data: Estimated Trend and Transitory Components

and Probabilities of Negative Shocks . . . 23 (a) KN Plucking Model: Real GDP and Its Trend Component . . . 23 (b) KN Plucking Model: Transitory Component . . . 23 (c) KN Plucking Model: Probabilities of Negative Shocks . . . 23 Figure 3.4 Australian Data: Estimated Trend and Transitory Components

and Probabilities of Negative Shocks . . . 24 (a) KN Plucking Model: Real GDP and Its Trend Component . . . 24 (b) KN Plucking Model: Transitory Component . . . 24 (c) KN Plucking Model: Probabilities of Negative Shocks . . . 24

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Figure 4.1 Forecasting U.S. GDP: A series of 16 realizations from the corre-sponding first-order Markov-process Stwas simulated starting at

time T + 1. The state ST was randomly drawn from a Bernoulli

distribution with probabilities P (ST = 0|ψT) and P (ST = 1|ψT)

taken from the corresponding Kim’s filter output. Subsequent states were simulated according to the corresponding transition matrix. . . 33 Figure 4.2 Forecasting Canadian GDP: A series of 16 realizations from the

corresponding first-order Markov-process Stwas simulated

start-ing at time T + 1. The state ST was randomly drawn from

a Bernoulli distribution with probabilities P (ST = 0|ψT) and

P (ST = 1|ψT) taken from the corresponding Kim’s filter output.

Subsequent states were simulated according to the corresponding transition matrix. . . 34 Figure 4.3 Forecasting Australian GDP: A series of 16 realizations from

the corresponding first-order Markov-process St was simulated

starting at time T + 1. The state ST was randomly drawn from

a Bernoulli distribution with probabilities P (ST = 0|ψT) and

P (ST = 1|ψT) taken from the corresponding Kim’s filter output.

Subsequent states were simulated according to the corresponding transition matrix. . . 35 Figure 4.4 1 quarter ahead forecasts of U.S. inflation and rolling RMSEs. . 40 (a) Inflation Forecasts . . . 40 (b) RMSE . . . 40 Figure 4.5 1 quarter ahead forecasts of Canadian inflation and rolling RMSEs. 41 (a) Inflation Forecasts . . . 41 (b) RMSE . . . 41 Figure 4.6 1 quarter ahead forecasts of Australian inflation and rolling

RM-SEs. The spike in inflation observed in the year 2000 is the result of Australia’s introduction of a goods and services tax. This spike in inflation results in poor forecasts of inflation. . . 42 (a) Inflation Forecasts . . . 42 (b) RMSE . . . 42

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ACKNOWLEDGEMENTS I would like to thank:

Dr. Graham Voss, for his support and encouragement throughout the process of preparing this thesis.

Dr. Ken Stewart and Dr. Glenn Otto, for helpful comments.

University of Victoria Department of Economics, for funding me with a Schol-arship.

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DEDICATION

This thesis is dedicated to Joanna who has always helped me and believed that I could do it.

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Introduction

The recent severe recession has brought new attention to the old debate in the business cycle literature on how variation in output should be allocated between permanent trend and transitory components. Commenting on the severity of the recession, Stock and Watson (2012) wrote:

The recession that began in the fourth quarter of 2007 was unprecedented in the postwar United States for its severity and duration. Following the NBER dated peak of 2007:4, GDP dropped by 5.5 percent and nearly 8.8 million jobs were lost. Based on the most recent revisions, the previous peak in GDP was not achieved for 15 quarters, in 2011:3, and as of this writing only 3.5 million jobs have been regained. All this suggests that the 2007:4 recession and recovery were qualitatively, as well as quantitatively, diﬀerent from previous postwar recessions.

Eﬀorts to describe the slow recovery diﬀer in whether they assume that trend is a smooth series that can be approximated by a trend line with transitory demand shocks responsible for most of the variation around the trend or that trend is a variable series subject to permanent shocks, i.e. supply shocks. The conventional view has been that variation in output is dominated by temporary deviations from trend and that the slow recovery is a slow reversion to trend. An alternative view, promoted by several authors, attributes the slow recovery to a slowdown in trend growth rates, see for example Stock and Watson (2012).

This debate on the decomposition of output into trend and transitory components has historically been expressed within a linear symmetric model framework. However, evidence, dating back as far as Friedman (1964), that output fluctuations are

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charac-terized by significant asymmetries and non-linearities suggests that questions on the decomposition of output could be better addressed in an asymmetric model frame-work. Friedman’s (1964) “plucking” model formalized by Kim and Nelson (1999) provides such an asymmetric model alternative.

The purpose of this thesis is to further our understanding of the role of permanent and transitory shocks and asymmetric output movements. This research is of interest both for describing historical GDP and for forecasting GDP. I propose an extension of the plucking model which allows for correlation between permanent and transitory innovations and test this model against Kim and Nelson’s (1999) plucking model and a linear symmetric model. Experiments indicate that the plucking models provide a good description of historical GDP and useful forecasts.

1.1 Business Cycle Asymmetry

Recent interest in business cycle asymmetry has brought new attention to the “pluck-ing” model suggested by Milton Friedman more than 40 years ago. Friedman (1964, 1993) noted that the amplitude of contractions in U.S. output tended to be strongly correlated with successive expansions, but that the amplitude of expansions were not correlated with successive contractions. This observed asymmetric serial correlation pattern led Friedman to propose his plucking model as an asymmetric alternative to the symmetric linear models often used to model business cycles.

In the plucking model, informally described by Friedman (1964, 1993), output is viewed as bumping along a ceiling of maximum feasible output, but it is plucked downward occasionally by a transitory contraction. The size of the transitory contrac-tion in output can vary widely. When subsequent recovery sets in, it tends to return output to the ceiling level. This model produces output series where the amplitude of contractions in output tend to be strongly correlated with successive expansions.

1.1.1 Formal Model

Kim and Nelson (1999) were the first to propose an econometric model of the busi-ness cycle that allows the decomposition of measures of economic activity into a trend component and deviations from the trend that exhibit the kinds of asymmetry identi-fied by Friedman. Their non-linear unobserved components (UC) model decomposes

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output into a trend ceiling component and a transitory component.

yt= τt+ ct, t = 1, . . . , T, (1.1)

where ytis the logarithm of observed output, τtis a trend ceiling and ctis a transitory

component. To allow for asymmetric deviations of output from the trend ceiling, the model assumes that the transitory component is subject to an asymmetric discrete shock which is dependent upon an unobserved variable St. Stis an indicator variable

that determines the nature of the shocks to the economy. When St= 0 the economy

is near the potential or trend output. When St= 1 the economy is hit by a transitory

negative shock. To account for a persistence of normal periods or recession periods, St is assumed to evolve according to a first-order Markov-switching process.

Kim and Nelson (1999) estimated the formal plucking model and tested it suc-cessfully against Clark’s (1987) linear symmetric UC model using quarterly real GDP data for the U.S. 1951:1-1995:3. Evidence in favour of the plucking model, or asym-metry in the transitory component of output, has also been reported by Mills and Wang (2002) and Simone and Clarke (2007). Mills and Wang (2002) applied Kim and Nelson’s methodology to the G-7 countries and found a fair degree of support for the plucking model. The results they reported for the U.S. 1951:1-2000:1 were very similar to those reported by Kim and Nelson (1999). More recently, Simone and Clarke (2007) estimated Kim and Nelson’s model and tested it against Clark’s (1987) model for 12 industrial and emerging economies. They found significant negative asymmetric shocks in all of the economies analyzed.

In Kim and Nelson’s (1999), Mills and Wang’s (2002) and Simone and Clarke’s (2007) analyses of U.S. data, their fitted models were consistent with a basic story of the business cycle where periods of normal growth are interrupted by sharp contrac-tions, followed by a period of rapid recovery. They also found evidence that during expansions output fluctuations are mainly permanent and that during recessions they are mainly transitory. According to these authors, the plucking model generally im-plies smooth permanent components and fluctuations that are driven primarily by transitory movements. This finding is consistent with the conventional view that variation in output is dominated by temporary deviations from trend and that the slow recovery is a slow reversion to trend.

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1.1.2 Correlated Innovations

The alternative view that the slow recovery is due to a slowdown in trend growth rates has found recent support in work by Morley et al. (2003). Morley et al. (2003) studied how variation is allocated between trend and transitory components in linear symmetric UC models. Estimating a UC model without the common zero-correlation restriction between permanent and transitory innovations for U.S. quarterly GDP, they found that the estimated trend component accounts for most of the variation in output. Furthermore, the zero-correlation restriction was rejected for U.S. quarterly GDP, with the estimated correlation being _{−0.9. This result is in contrast to the} findings of Clark (1987) based on estimates of zero-correlation UC models, which suggest that fluctuations in output are primarily transitory.

Morley et al.’s (2003) result for linear models provides a persuasive reason to gen-eralize the Kim and Nelson (1999) plucking model to allow for correlated innovations. Generally, there is no reason to assume that the permanent and transitory compo-nent are uncorrelated. The importance of permacompo-nent shocks will be confirmed, if permanent innovations are found to be strongly negatively correlated with transitory innovations.

1.2 Structure of the Thesis

In this thesis, I present a more general plucking model of the business cycle that allows for asymmetric shocks, as well as the possibility that permanent and transi-tory innovations are correlated. Extending the plucking model to allow for correlated innovations addresses a key concern about the role of permanent movements in the business cycle. My results suggest that asymmetry is important in explaining business cycle dynamics in the three countries studied, the U.S., Canada and Australia. Cor-related innovations are important for the the U.S. but not for Australia or Canada. Specifically, the coeﬃcient associated with the discrete shock in the transitory com-ponent is negative and significant in all of the economies analyzed and the restriction that innovations in the unobserved trend and cycle are uncorrelated was rejected for the U.S..

The remainder of this thesis is organized as follows: Chapter 2 describes the gen-eralized plucking model and shows that it nests Kim and Nelson’s (1999) plucking model. I am not the first to propose UC models with correlated innovations and

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asymmetric shocks. A recent paper (Sinclair, 2010) considered the possibility. Her approach diﬀers from the one describe here, in that she starts with Morley et al.’s (2003) linear UC model with correlated innovations and generalizes it to allow for asymmetric shocks. As such, her model does not nest the plucking model and there-fore likelihood ratio tests are not available for testing the zero-correlation restriction. In Chapter 3 the generalized plucking model is applied to real GDP series of the U.S., Canada and Australia and tested against both Kim and Nelson’s (1999) plucking model and Clark’s (1987) linear symmetric model. Chapter 4 studies the forecast-ing performance of the pluckforecast-ing model. Chapter 5 draws conclusions from results of Chapters 3 and 4.

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Chapter 2 The Model

The correlated innovations plucking model presented in this chapter extends Kim and Nelson’s (1999) plucking model to allow for correlated trend and cycle innovations in the spirit of Morley et al. (2003). The key features of this model are that it allows for asymmetry in the transitory component via a Markov-switching process and at the same time it allows for correlation between innovations within the model. As pointed out by Morley and Piger (2012), transitory innovations could be due to the same factors that drive permanent innovations or they could be due to independent factors. Therefore, it is best not to assume independence in advance.

2.1 The Model

Following Kim and Nelson (1999), the logarithm of output, yt, is decomposed into

two unobserved components:

yt= τt+ ct (2.1)

where τt represents the permanent (or trend) component and ct represents the

tran-sitory component.

The transitory component is modelled as a second order autoregressive AR(2) process, where shocks to the process are assumed to be a mixture of both a symmetric and an asymmetric shock. This asymmetric shock captures the “plucks” of Friedman’s plucking model. The transitory component and the shocks aﬀecting its behaviour are

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specified as follows: ct = φ1ct−1+ φ2ct−2+ γSt+ ut (2.2) ut ∼ N(0, σ2u,St) (2.3) σu,S2 t = σ 2 u,0(1− St) + σu,12 St (2.4) St = 0 or 1 (2.5)

In the above specification, the term ut is the usual symmetric shock. The term

γStis an asymmetric discrete shock, which is dependent upon an unobserved variable

denoted by St, which is an indicator variable that determines the nature of the shocks

to the economy. When the economy is near the trend output, it is said to be in a normal state. In this case, St= 0, which implies that γSt= 0. In a state of recession,

the economy is hit by a transitory shock with a negative expected value, that is, γSt = γ < 0. Temporary disturbances are plucking output down below its trend

ceiling level. In the absence of such shocks, ct evolves as an AR(2) process. The

variance of the symmetric shock may be diﬀerent during normal and recession times and equation (2.4) allows the variance to depend on St. For example, if the trend

represents an absolute ceiling level, then σ2

u,0 should be equal to zero.

The permanent component is modelled as a random walk. Friedman (1993) sug-gested that the permanent component, or what he named “the ceiling maximum feasible output”, could be approximated by a pure random walk, with various types of shocks producing disturbances to it. The permanent component is written as:

τt = gt−1+ τt−1+ vt (2.6)

gt = gt−1+ wt (2.7)

wt ∼ N(0, σw2) (2.8)

vt ∼ N(0, σv,S2 t) (2.9)

σ_v,S2 _t = σ_v,02 (1_{− S}t) + σ2v,1St (2.10)

where, the permanent component τt is subject to shocks to the level, vt, and shocks

to the growth rate, wt. In some implementations the growth rate, gt, is assumed

constant, but as mentioned by Clark (1987), it is not appropriate to assume a constant growth in advance. For example, the decline of U.S. productivity growth in the 1970s and the reduction of labour force growth in the 1980s are likely to have reduced the

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growth rate of output. Modelling gt to be non-constant, allows for the possibility of

changes in the growth rate of output. Note that the above specification again allows for the variance of the shock to the level to be diﬀerent during normal and recession times. However, variance of the shock to the growth rate is the same during normal and recession times.

In order to account for a persistence of normal periods or periods of recession, it is assumed that St evolves according to a first-order Markov-switching process as in

Hamilton (1989): That is

P (St = 1|St−1 = 1) = p (2.11)

and

P (St= 0|St−1 = 0) = q. (2.12)

The state of the economy (whether St = 0 or 1) is thus determined endogenously in

the model.

Notice that the plucking model nests the linear symmetric model suggested by Clark (1987). The model of Clark (1987) does not account for asymmetries, which in the context of the plucking model implies that γ = 0 and σv0 = σv1 = σv and

σu0= σu1= σu.

2.1.1 Extended Model

To extend Kim and Nelson’s (1999) model, the innovations (vt and ut) in equations

(2.2) and (4.4) are assumed to be jointly normally distributed random variables with mean zero and general covariance matrix,

Σ = � σ2 v,St σvu,St σuv,St σ 2 u,St � , (2.13)

which allows for correlation between τt and ct. Of course, the model of Kim and

Nelson (1999) is nested as a special case of this model with zero correlation between ut and vt.

In the next chapter, the correlated innovations plucking model is tested against Kim and Nelson’s (1999) plucking model and the linear symmetric model proposed by Clark (1987). Rejection of the linear symmetric model indicates the presence of asymmetries and rejection of Kim and Nelson’s (1999) plucking model indicates the presence of correlated innovations or some other mis-specification.

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2.1.2 Identification

An identification problem can arise when fitting UC models of which the correlated innovation plucking model is a special type. The problem concerns the possibility of there being insuﬃcient information to distinguish between model parameters. Given a model with likelihood function f (y|θ) and parameter vector θ ∈ Rk_{, if the two values}

θ1 and θ2 are observationally equivalent, so that f (y|θ1) = f (y|θ2), then the model

is globally unidentified. More than one set of values for the parameters gives rise to identical values of the likelihood function and the data gives no guide for choosing among maximum likelihood estimates (MLEs).

In plucking models global identification of the state variable can be achieved by restricting the sign of the discrete, asymmetric innovation, γ, to be non-positive (Sinclair, 2010). However, there is little guidance on which restrictions are required for global identification of the variance parameters. Therefore, in order to ensure that MLEs are unique, I investigated the lesser condition of local identifiability of the model at the MLE point. A parameter vector θ is said to be locally identifiable if there exists an open neighbourhood of θ containing no other θ which is observationally equivalent (McDonald, 1982). Rothenberg (1971) showed that a model is locally identified at θ if, and only if, the rank of the information matrix is equal to the number of free parameters in the model (equivalently, the information matrix is nonsingular in a neighbourhood around θ). A common symptom of trying to estimate an unidentified model is diﬃculty with inverting the information matrix and computing estimated standard errors. Hence, local identifiability can be investigated by inspecting the estimated standard errors of the MLEs.

2.1.3 Estimation

This section describes estimation of the plucking models based on Kim’s (1993) ap-proximate MLE. Kim’s filter may be considered a combination of extended versions of the Kalman filter and the Hamilton filter, along with appropriate approximations. In order to apply Kim’s filter, the model must first be written in state-space form. The state-space model consists of two equations: an observation equation and a state equation. The observation equation describes the relation between observed variables and unobserved state variables. The state equation describes the dynamics of the state variable. The state equation has the form of a first order diﬀerence equation in the state vector. Writing the plucking model in state-space form, the observation

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and state equations are yt= hTξt (2.14) ξ_t= µ_S_t+ Fξ_t₋₁+ vt (2.15) vt∼ N(0, QSt) (2.16) where h =       1 1 0 0      , ξt =       τt ct ct−1 gt      , µSt =       0 γSt 0 0      , F =       1 0 0 1 0 φ1 φ2 0 0 1 0 0 0 0 0 1      , vt =       vt ut 0 wt      . (2.17) The correlated innovations plucking model has covariance matrix

QSt =       σ2 v,St σvu,St 0 0 σuv,St σ 2 u,St 0 0 0 0 0 0 0 0 0 σ2 w      . (2.18)

Kim and Nelson’s (1999) plucking model has covariance matrix

QSt =       σ2 v,St 0 0 0 0 σ2 u,St 0 0 0 0 0 0 0 0 0 σ2 w      . (2.19)

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can be written as follows:

ξ(i,j)_t_|t−1 = µj + Fξit−1|t−1, (2.20)

P(i,j)_t_|t−1 = FPi_t_−1|t−1FT + Qj, (2.21)

η_t|t−1(i,j) = yt− hξ(i,j)_t|t−1, (2.22)

f_t(i,j)_|t−1 = hP(i,j)_t_|t−1hT, (2.23)

ξ(i,j)_t|t = ξ(i,j)_t|t−1+ P(i,j)_t|t−1hT[f_t|t−1(i,j)]−1η(i,j)_t|t−1, (2.24) P(i,j)_t_|t = (I_{− P}(i,j)_t_|t−1hT[f_t(i,j)_|t−1]−1)hP(i,j)_t_|t−1, (2.25)

ξj_t|t = �2

i=1P rob[St−1 = i, St= j|Ψ]ξ(i,j)_t_|t

where ξ(i,j)_t|t−1 is an inference about ξt based on information up to time t− 1, ξ (i,j) t|t is an

based on information up to time t_{− 1, f}_t(i,j)_|t−1 is the variance of η_t(i,j)_|t−1 and Ψt refers to

information available at time t.

At each iteration of the algorithm, the conditional density of yt is given by

The log likelihood function can be obtained by summing equation (2.28) over time as

LL =

T

�

t=1

log(f (yt|Ψt−1)). (2.30)

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Chapter 3 Experiments

The correlated innovations plucking model presented in Chapter 2 was fit to the logarithm of quarterly GDP of three developed countries: the U.S., Canada and Australia. The three data series are seasonally-adjusted quarterly U.S. GDP for the sample period 1947:Q1 to 2011:Q1 taken from the FRED database, seasonally-adjusted quarterly Canadian GDP for the sample period 1961:Q1 to 2011:Q1 taken from the CANSIM database and seasonally-adjusted quarterly Australian GDP for the sample period 1959:Q3 to 2011:Q1 taken from the Australian Bureau of Statistics National Accounts database.

Estimation of plucking model parameters was performed using Kim’s (1993) filter. The filter was implemented in R code (provided in the appendix) and Nelder-Mead numerical optimization was used to maximize the likelihood function derived from the filter.

3.1 Model Selection

Tables 3.1, 3.2 and 3.3 summarize the model fitting results for the U.S., Canada, and Australia (values in parentheses indicate the standard errors). Model 1 is the corre-lated innovations plucking model with asymmetry in the transitory component and correlated innovations. Model 2 is Kim and Nelson’s plucking model which restricts σuv0 = σuv1 = 0. Kim and Nelson’s model was also estimated restricting the growth

rate variance to be zero, σ2

w = 0. This model will be referred to as Model 3. Kim and

Nelson’s model was also estimated restricting symmetric transitory innovation vari-ances to be zero, σ2

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model will be referred to as Model 4. Finally, Clark’s (1987) linear symmetric model, referred to as Model 5, was estimated in order to test the importance of asymmetric shocks.

Looking first at the results for the U.S., the correlated innovations plucking model (model 1) appears to represent an improvement over both Kim and Nelson’s plucking model (model 2) and the linear symmetric model (model 5). No diﬃculties were en-countered inverting the information matrix to compute the estimated standard errors for the correlated innovations plucking indicating that the model is at least locally identified at the MLE. Testing the restriction of a linear symmetric model, i.e. that γ = 0, by comparing model 2 and model 5 the likelihood ratio test statistic is 34.9170, rejecting the null hypothesis at an approximate 0.0000 significance level. Asymmetry is indeed important for explaining the movements in U.S. GDP. Testing the zero-correlation restriction of Kim and Nelson’s plucking model, i.e. that σuν0 = σuν1 = 0,

by comparing model 1 and model 2 the likelihood ratio test statistic is 6.4535, re-jecting the null hypothesis at a 0.0397 significance level. Correlated innovations are important for explaining movements in U.S. GDP. The estimated correlation coeﬃ-cients for normal and recession states are −0.5343 and 0.9990, respectively. During normal times permanent and transitory innovations are negatively correlated and tend to move in opposite directions.

Consider the permanent component of U.S. GDP. The permanent component is subject to shocks to the level, vt, and shocks to the growth rate, wt. Testing

the hypothesis that the trend growth has been constant is equivalent to testing the restriction that σw = 0. A likelihood ratio test was not available to test this restriction

due to diﬃculties encountered fitting the restricted model, instead a Wald test was performed. The Wald test statistic is 2.23202, rejecting the null hypothesis at a 0.0676 significance level. The trend component is subject to both types of shocks.

Consider the transitory component of U.S. GDP. The transitory component is subject to two diﬀerent shocks: an asymmetric, discrete, shock πSt and a symmetric,

continuous, shock ut. The parameters σu0 and σu1measure the relative importance of

the symmetric shock. A test of the joint hypothesis that σu0= σu1= 0 was performed

using the Wald test statistic. A likelihood ratio test would have been preferred but again due to diﬃculties fitting the restricted model such a test was not feasible. The Wald statistic for the joint hypothesis that σ2

u,1 = σu,02 = 0 is 6.5773, rejecting the

null at a 0.0187 significance level. This test result suggests that both symmetric and asymmetric shocks are important for explaining the dynamics in the transitory

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component. The significant shock means that the trend cannot be interpreted as an absolute maximum ceiling value. At times, positive symmetric transitory shocks may cause output to exceed the trend value.

Looking at Canada, model 1 does not appear to represent an improvement over model 2. Testing σuν0 = σuν1 = 0, the likelihood ratio test statistic is 0.5369, failing

to reject the null hypothesis at a 0.9907 significance level. Model 2 does, however, represent an improvement over model 5. The LR statistic for the hypothesis γ = 0 is 13.30979. Comparing models 2 and 3, the LR statistic for the hypothesis σw = 0

is 10.2547, rejecting the null at significance level 0.0014. Canadian trend growth has not been constant. The LR statistic for the hypothesis that σu0 = σu1= 0 is 2.6628,

failing to reject the null at a 0.2641 significance level. The trend ceiling hypothesis is not rejected for Canada.

Looking at Australia, model 1 does not appear to represent an improvement over Kim and Nelson’s model, model 2. Testing σuν0 = σuν1= 0, the likelihood ratio test

statistic is 0.0313, failing to reject the null hypothesis at a 0.9845 significance level. The evidence against the symmetric model, however, is strong. Comparing model 2 with model 5 the LR statistic for the hypothesis γ = 0 is 6.6169. Wald tests were used for testing σw = 0 and σu0 = σu1 = 0 due to failures of convergence in fitting

models 3 and 4. The Wald test statistic for the hypothesis σw = 0 is 2.6356, failing to

reject the null at a 0.1045 significance level. The Wald test statistic for the hypothesis σu0= σu1= 0 is 8.4384, rejecting the null at a 0.0147 significance level.

In summary, the correlated innovations plucking model has been selected for the U.S. and Kim and Nelson’s plucking model has been selected for Canada and Aus-tralia. The symmetric model has been rejected for all three countries, suggesting that the discrete asymmetric shock γSt in addition to traditional symmetric shocks to the

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Parameters Model 1 Model 2 Model 3 Model 4 Model 5 (Correlated) (KN Plucking) (σw= 0) (σu0= σu1= 0) (Symmetric)

p 0.9572 0.9317 0.9280 - -(0.0519) (0.0420) (0.0537) - -q 0.9692 0.9648 0.9609 - -(0.0309) (0.0236) (0.0261) - -φ1 1.4292 1.5149 1.5936 - 1.5204 (0.0752) (0.1369) (0.1422) - (0.1042) φ2 -0.6392 -0.6594 -0.6298 - -0.5824 (0.0921) (0.1171) (0.1385) - (0.1044) σu0 0.0033 0.0020 0.0028 - 0.0061 (0.0013) (0.0009) (0.0008) - (0.0011) σu1 0.0025 0.0000 0.0000 - -(0.0015) - - - -σν0 0.0051 0.0043 0.0042 - 0.0055 (0.0016) (0.0005) (0.0006) - (0.0011) σν1 0.0097 0.0118 0.0127 - -(0.0004) (0.0011) (0.0012) - -σw 0.0006 0.0007 - - 0.0002 (0.0004) (0.0003) - - (0.0001) γ -0.0047 -0.0062 -0.0041 - -(0.0032) (0.0019) (0.0017) - -σuν0 -8.9184e-06 - - - -(1.3208e-05) - - - -σuν1 2.3891e-05 - - - -(1.3071e-05) - - - -Log Likelihood 806.7742 803.5474 803.0858 - 786.0889

Table 3.1: U.S. Data: Maximum Likelihood Estimation of the Models (Model 2 and 3 ML estimates of σu1 fell on the boundary (σu1=0). To calculate standard errors σu1

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p 0.7223 0.7077 0.8066 0.7727 -(0.1271) (0.0928) (0.2174) (0.1225) -q 0.9835 0.9837 0.9846 0.9858 -(0.0185) (0.0122) (0.0186) (0.0135) -φ1 1.2248 1.3013 1.2913 1.2460 1.4757 (0.8068) (0.1967) (0.1455) (0.2395) (0.1966) φ2 -0.3074 -0.3747 -0.3222 -0.3244 -0.5397 (0.7457) (0.1935) (0.1437) (0.2066) (0.1923) σu0 0.0042 0.0042 0.0058 - 0.0053 (0.0156) (0.0027) (0.0014) - (0.0018) σu1 0.0000 0.0000 0.0001 - -- - (0.0012) - -σν0 0.0049 0.0052 0.0041 0.0068 0.0048 (0.0124) (0.0023) (0.0023) (0.0004) (0.0017) σν1 0.0032 0.0035 0.0028 0.0032 -(0.0014) (0.0010) (0.0018) (0.0014) -σw 0.0005 0.0004 - 0.0005 0.0005 (0.0003) (0.0002) - (0.0002) (0.0003) γ -0.0148 -0.0135 -0.0118 -0.0147 -(0.0086) (0.0029) (0.0025) (0.0030) -σuν0 1.2338e-06 - - - -(1.0780e-05) - - - -σuν1 1.6719e-06 - - - -(1.68413-06) - - - -Log Likelihood 628.4773 628.2088 623.0815 626.8774 621.5539

Table 3.2: Canadian Data: Maximum Likelihood Estimation of the Models (Model 1 and 2 ML estimates of σu1 fell on the boundary (σu1=0). To calculate standard

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p 0.7370 0.7370 - - -(0.1181) (0.0876) - - -q 0.9278 0.9278 - - -(0.0292) (0.0260) - - -φ1 0.6553 0.6553 - - 1.4392 (0.1183) (0.0948) - - (0.0306) φ2 0.0328 0.0328 - - -0.9344 (0.0817) (0.0724) - - (0.0372) σu0 0.0075 0.0075 - - 0.0002 (0.0028) (0.0026) - - (0.0011) σu1 0.0002 0.0000 - - -(0.0048) (0.0039) - - -σν0 0.0043 0.0043 - - 0.0116 (0.0117) (0.0060) - - (0.0007) σν1 0.0074 0.0074 - - -(0.0050) (0.0013) - - -σw 0.0006 0.0006 - - 0.0001 (0.0007) (0.0004) - - (0.0006) γ -0.0213 -0.0213 - - -(0.0022) (0.0021) - - -σuν0 -3.7491e-07 - - - -(8.4674e-06) - - - -σuν1 -1.1734e-06 - - - -(3.3835e-05) - - - -Log Likelihood 569.1627 569.147 - - 565.8386

Table 3.3: Australian Data: Maximum Likelihood Estimation of the Models (Model 3 ML estimate of σu1 fell on the boundary (σu1=0). To calculate standard errors σu1

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3.2 Interpretation of Parameter Estimates

The variance parameters presented in Table 3.1 show how volatility in U.S. GDP is allocated between permanent and transitory components. In the normal state, volatility in the permanent component is described by the variance parameters σν0

and σw. Table 3.1 shows that the variance parameter σν0 of model 1 is approximately

20% larger than that of model 2, indicating increased volatility in the permanent component. This result is consistent with Morley et al.’s (2003) finding, that models without correlation imply a very smooth trend and a cycle that is large in ampli-tude and highly persistent, whereas, models with correlation imply that much of the variation in GDP is variation in trend.

Probabilities and durations of normal and recession states for the U.S. correlated innovations plucking model can be derived from the transition probability parameter estimates. Based on the estimates ˆp = 0.9572 and ˆq = .9692 the expected duration of a recession (State 1) is 1/(1_{− ˆp) = 23.3645 and of normal times (State 0) is} 1/(1− ˆq) = 32.4675. The steady state probabilities of being in a recession or normal state are P (St = 1) = 1− ˆq 2_{− ˆp − ˆq}= .4185 and P (St= 0) = 1− ˆp 2_{− ˆp − ˆq}= .5815, (3.1) respectively. Most of the time, the U.S. is in the normal state (St = 0), subject

principally to permanent disturbances and operating near the trend ceiling. The onset of a recession produces a sequence of negative transitory shocks which pluck output down. Note that ˆγ =−0.0047 and is clearly significantly negative, suggesting strong support for the plucking model. The sum of the autoregressive cycle coeﬃcients, φ1 + φ2, is 0.7900 suggesting that once the sequence of negative shocks end, their

eﬀects decay rapidly.

Estimated parameters from the U.S. Kim and Nelson’s plucking model are con-sistent with Kim and Nelson (1999) and so interpretation of the parameters is worth repeating. Based on the estimates ˆp = 0.9317 and ˆq = .9648 the expected dura-tion of a recession (State 1) is 1/(1− ˆp) = 14.6413 and of normal times (State 0) is 1/(1− ˆq) = 28.4091. The steady state probabilities of being in a recession or normal state are P (St = 1) = 1− ˆq 2_{− ˆp − ˆq}= .3401 and P (St= 0) = 1− ˆp 2_{− ˆp − ˆq}= .6599, (3.2)

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respectively. Note that ˆγ = −0.0062 and is clearly significantly negative, suggesting strong support for the plucking model. The sum of the autoregressive cycle coeﬃ-cients, φ1+ φ2, is 0.8555 suggesting that once the sequence of negative shocks end,

their eﬀects decay rapidly.

For Canada the estimates ˆq and ˆp imply expected durations of 3.4211 and 61.3497 quarters for recession and normal times respectively, with steady state probabilities of 0.0528 and 0.9472, respectively. The estimate of the impact of the negative transitory shocks is ˆγ = −0.0135, which is again negative but rather larger than for the U.S.. The sum of the AR coeﬃcients is φ1 + φ2 = 0.9266, a little higher than for the U.S.

This result again suggests that there are no subsequent eﬀects on the series once the sequence of negative shocks end.

For Australia the estimates ˆq and ˆp imply expected durations of 3.8023 and 13.8504 quarters for recession and normal times respectively, with steady state probabilities of 0.2154 and 0.7846, respectively. ˆγ = _{−0.0213 and is significantly negative. The} sum of the AR coeﬃcients is φ1+ φ2 = .6881, implying that the negative shocks are

somewhat longer lived than for the U.S. or Canada.

In conclusion, all three economies are subject to negative transitory shocks, which pluck output down. Near the end of a recession, with no further new negative shocks, the eﬀects of negative shocks decays rapidly and the economies return to the trend ceiling.

3.3 Permanent and Transitory Components

Figures 3.1, 3.3 and 3.4 present the filtered estimates of the unobserved components of selected plucking models and probabilities of being in a recession for the U.S., Canada and Australia. Kim and Nelson (1999) provide details of the construction of estimates: the first 20 quarters of each sample are used to obtain initial values for the filter and these observations are thus omitted from the plots.

Figure 3.1(a) presents the logarithm of U.S. GDP, yt, and a filtered estimate

of its trend component, τt. The figure shows that most of the time the U.S. is

operating below the trend ceiling. Figure 3.1(b) presents a filtered estimate of the transitory component, ct. The transitory component appears to move in general

with the business cycle, as indicated by the shaded NBER recession dates. During the short periods right after the NBER recessionary periods, the negative transitory shocks are deteriorating restoring the economy back to the trend ceiling. Figure 3.1(c)

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presents the probabilities of negative shocks to the transitory component. There is positive probability of a transitory asymmetric shock for all of the NBER dated recessions, with 7 of the 9 recessions in the sample having probability near 1. For the recessions with a high probability of transitory asymmetric shocks the series drops below the permanent component. These recessions have the appearance of a pluck as described by Friedman such that the permanent component appears to be a ceiling and the series is temporarily plucked away from that ceiling. The two recessions where the probability of an asymmetric transitory shock remains low are 1990:3-1991:1, and 2001:1-2001:4. For these recessions, the reduction in output is largely permanent, as can be seen in Figure 3.1(a). There is a peak-to-trough movement in the transitory component, but it is smaller in general than in the recessions that experienced asymmetric shocks. These recessions appear to have diﬀerent features than the pluck recessions.

Figures 3.3(a) and 3.3(b) show that Canadian output has been close to its ceil-ing value except for three major episodes - durceil-ing the early 1980s, durceil-ing the early 1990s and during the 2007:4 recession. The onset of these three episodes correspond to recessions found in the business cycle chronology for Canada, and are seen to be a consequence of three major negative shocks to the transitory component of out-put. Figure 3.3(c) presents the probabilities of negative shocks to the transitory component. The 1980s recession is identified with a single spike in the probability of asymmetric shocks whereas the 1990s recession corresponds to a period with several quarters with high probability of asymmetric shocks. Both recessions were relatively long in duration. Recessions in Canada are not exactly coincident with those experi-enced in the US but rather they always occur after a recession in the U.S..

Figures 3.4(a), 3.4(b) and 3.4(c) show that, unlike the U.S. and Canada, Australia achieved a long period of normal growth starting in the mid 1990s. Recessionary periods are again closely associated with large negative values of, and the probability of negative shocks to, the transitory component. These periods shown as shaded areas are constructed from the business cycle chronologies. Although rare, the asymmetric shocks appear important.

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Time log GDP 1950 1960 1970 1980 1990 2000 2010 8.0 8.5 9.0 9.5 Actual Output Trend

(a) Correlated Plucking Model: Real GDP and Its Trend Component

Time log GDP 1950 1960 1970 1980 1990 2000 2010 − 0.05 − 0.04 − 0.03 − 0.02 − 0.01 0.00 0.01

(b) Correlated Plucking Model: Transitory Component Time Probability 1950 1960 1970 1980 1990 2000 2010 0.0 0.2 0.4 0.6 0.8 1.0

(c) Correlated Plucking Model: Probabilities of Negative Shocks

Figure 3.1: U.S. Data: Estimated Trend and Transitory Components and Probabili-ties of Negative Shocks for the Correlated Plucking Model

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Time log GDP 1950 1960 1970 1980 1990 2000 2010 8.0 8.5 9.0 9.5 Actual Output Trend

(a) KN Plucking Model: Real GDP and Its Trend Component

Time log GDP 1950 1960 1970 1980 1990 2000 2010 − 0.05 − 0.04 − 0.03 − 0.02 − 0.01 0.00 0.01

(b) KN Plucking Model: Transitory Component

Time Probability 1950 1960 1970 1980 1990 2000 2010 0.0 0.2 0.4 0.6 0.8 1.0

(c) KN Plucking Model: Probabilities of Nega-tive Shocks

Figure 3.2: U.S. Data: Estimated Trend and Transitory Components and Probabili-ties of Negative Shocks for Kim and Nelson’s Plucking Model

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Time log GDP 1970 1980 1990 2000 2010 26.6 26.8 27.0 27.2 27.4 27.6 27.8 Actual Output Trend

Time log GDP 1970 1980 1990 2000 2010 − 0.08 − 0.06 − 0.04 − 0.02 0.00

Time Probability 1970 1980 1990 2000 2010 0.0 0.2 0.4 0.6 0.8 1.0

Figure 3.3: Canadian Data: Estimated Trend and Transitory Components and Prob-abilities of Negative Shocks

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Time log GDP 1970 1980 1990 2000 2010 11.0 11.5 12.0 12.5 Actual Output Trend

Time log GDP 1970 1980 1990 2000 2010 − 0.06 − 0.05 − 0.04 − 0.03 − 0.02 − 0.01 0.00

Time Probability 1970 1980 1990 2000 2010 0.0 0.2 0.4 0.6 0.8 1.0

Figure 3.4: Australian Data: Estimated Trend and Transitory Components and Prob-abilities of Negative Shocks

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3.4 The Great Moderation

The idea of the “Great Moderation” was made popular by Ben Bernanke (then mem-ber and now chairman of the Board of Governors of the Federal Reserve) in a speech at the 2004 meeting of the Eastern Economic Association. He stated that “one of the most striking features of the economic landscape over the past twenty years or so has been a substantial decline in macroeconomic volatility.”

Kim and Nelson (1999b) and McConnell and P´erez-Quiros (2000) were the first to identify this moderation in the volatility of U.S. GDP growth. Both papers presented evidence of a large reduction in the volatility of U.S. GDP growth and estimated it to have occurred in the first quarter of 1984. Stock and Watson (2002) tried to characterizes this decline in volatility oﬀering possible factors to explain it, such as, better technology, better policy and good luck.

The severity of the 2007:4 recession has led to claims that the Great Moderation is now over or was even a myth in the first place. However, Morley (2009) argued that the stabilization of economic activity since the mid-1980s was very much a reality and that it is likely to continue.

Table 3.4 presents the estimates from fitting plucking and symmetric UC models separately to U.S. quarterly GDP for the periods 1951:1-1983:4 and 1984:1-2011:1 to capture the decline in volatility. Model 1 is the correlated innovations plucking model. Model 2 is Kim and Nelson’s plucking model. Model 3 is Clark’s (1987) linear symmetric model. Likelihood ratio tests selected Kim and Nelson’s plucking model in both time periods. The estimates in columns pre 1984 and post 1983 show the change in the standard deviation of the permanent innovation (σν0 and σν1) and the standard

deviation of the transitory innovation (σu0and σu1). Reduction in volatility is evident

in both permanent and transitory innovations in both normal and recession states. For the normal state, the post 1983 standard deviation of the transitory innovation, σu0= 0.0015 is less than 40% of the pre 1984 standard deviation, σu0 = 0.0042. The

post 1983 standard deviation of the permanent innovation, σν0 = 0.0042, is less than

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pre 1984 post 1983

Par Model 1 Model 2 Model 3 Model 1 Model 2 Model 3 (Correlated) (KN Plucking) (Symmetric) (Correlated) (KN Plucking) (Symmetric)

p 0.6524 0.6868 - 0.2468 0.2776 -(0.2410) (0.1772) - (0.2556) (0.2679) -q 0.8922 0.9037 - 0.9702 0.9758 -(0.0399) (0.0733) - (0.0179) (0.0164) -φ1 1.3496 1.3575 1.4777 1.5248 1.5677 1.6316 (0.1186) (0.1575) (0.1349) (0.0624) (0.0817) (0.1419) φ2 -0.6535 -0.5760 -0.6039 -0.5533 -0.5972 -0.7044 (0.1555) (0.2135) (0.1698) (0.0641) (0.0796) (0.1384) σu0 0.0061 0.0042 0.0069 0.0015 0.0015 0.0032 (0.0026) (0.0042) (0.0025) (0.0009) (0.0007) (0.0009) σu1 0.0024 0.0000 - 0.0007 0.0000 -(0.0039) (0.0048) - (0.0012) (0.0008) -σν0 0.0114 0.0072 0.0073 0.0040 0.0042 0.0036 (0.0021) (0.0023) (0.0020) (0.0009) (0.0005) (0.0007) σν1 0.0087 0.0112 - 0.0007 0.0013 -(0.0042) (0.0024) - (0.0006) (0.0013) -σw 0.0002 0.0003 0.0003 0.0004 0.0004 0.0007 (0.0003) (0.0003) (0.0004) (0.0003) (0.0003) (0.0004) γ -0.0088 -0.0097 - -0.0147 -0.0149 -(0.0027) (0.0036) - (0.0012) (0.0014) -σuν0 -0.0001 - - 0.0000 - -(0.0000) - - (0.0000) - -σuν1 0.0000 - - -0.0000 - -(0.0000) - - (0.0000) - -LL 400.2032 399.1498 397.3378 346.1728 344.6556 337.4798

Table 3.4: Maximum Likelihood Estimation of the Models for Time Periods 1947:1-1983:4 and 1984:1-2011:1

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Chapter 4 Forecasting

Chapter 3, showed the usefulness of the plucking model for describing historical GDP. The model captured the distinctive features of expansions and recessions and through Markov switching predicted the switch from expansion to recession and vice versa. This chapter shows how these features make the model useful for forecasting. First, a direct application of the model to forecast log GDP is described and then application of an activity measure, derived from the model, to forecasting inflation is described.

4.1 Forecasting GDP

This section discusses forecasting log GDP s-steps ahead, yT +s. Let ˆyT +s|T denote a

forecast of yT +s based on y1, . . . , yT. Economists conventionally use linear statistical

models for forecasting. These models imply that all characteristics of the variable under analysis are constant over time. Consequently, in linear models, the underly-ing properties of variables analyzed do not diﬀer between recessions and expansions. Though linear models have been reasonably successful as a practical tool for fore-casting, they are inherently limited in the presence of non-linearities in data and consequently forecasts from them could be misleading. In view of these limitations plucking models are expected to outperform linear models for forecasting.

4.1.1 Forecasting with Linear Models

Historically, forecasts based on Clark’s (1987) standard linear model have served as a useful benchmark and I describe them here before turning to the plucking model

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based forecasts. Clark’s linear model is given by: yt= τt+ ct (4.1) ct= φ1ct−1+ φ2ct−2+ ut (4.2) ut∼ N(0, σ2u) (4.3) τt= gt−1+ τt−1+ vt (4.4) gt= gt−1+ wt (4.5) wt∼ N(0, σ2w) (4.6) vt∼ N(0, σ2v) (4.7)

This model can be written in state space form following as

yt= hTξt (4.8)

ξt= Fξt−1+ vt (4.9)

vt∼ N(0, Q) (4.10)

where h, ξ_t, F and vt are as in (2.17) and

Q =       σ2 v 0 0 0 0 σ2 u 0 0 0 0 0 0 0 0 0 σ2 w      . (4.11)

Once the system is written in state-space form it is easy to use the Kalman Filter to calculate exact finite sample s-period ahead forecasts, ˆyT +s|T. The calculation is

described below, see Hamilton (1994) for a more detailed description.

Observe that the state equation (4.9) has the form of a first order diﬀerence equation in the state vector ξt and that this equation is easy to solve by recursive

substitution,

ξt+s = Fsξt+ Fs−1vt+1+ F1vt+s−1+ vt+s. (4.12)

Future values of the state vector depend on (ξt, ξt−1, . . .) only through the current

value ξ_t. It follows that, the minimum mse projection of ξ_t+s on ξ_tand ytis given by

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From the law of iterated projections, the s-step ahead forecast of the state vector is ˆ

ξT +s|T = FsξˆT|T (4.14)

with MSE matrix

PT +s|T = FsPT|T(FT)s+ Fs−1Q(FT)s−1+· · · + FQFT + Q, (4.15)

where ˆξ_T_|T is an inference on ξ_T conditional on yT and PT|T is the MSE associated

with this linear projection. Values of ˆξ_T_|T and PT|T are available from the final

iteration of the Kalman Filter applied to y1, . . . , yT.

To forecast the observed vector yT +s recall from the observation equation (4.8)

that

yT +s = hTξT +s. (4.16)

The s-period ahead forecast of y is then ˆ

yT +s|T = hTξˆT +s|T = hTFsξˆT|T, (4.17)

with MSE matrix

M SE(ˆyT +s|T) = E[(yT +s− ˆyT +s|T)(yT +s− ˆyT +s|T)T] = hTPT +s|Th. (4.18)

Assuming that future prediction errors are normally distributed, prediction inter-vals for yT +s|T are given by

� ˆ yT +s|T − z1−α/2 � M SE(ˆyT +s|T), ˆyT +s|T + z1−α/2 � M SE(ˆyT +s|T) � , (4.19) see for example Durbin and Koopman (2001).

Forecasts of yT +s for s = 1, . . . , 16 and prediction intervals based on the T most

recent values for the U.S., Canada and Australia are presented in Figure ??. Forecasts show positive future growth following the historical trend rate of growth centred within symmetric upper and lower prediction interval limits. The linear model cannot account for the possibility of an economy falling into recession and is dominated by the many quarters of positive historical growth.

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4.1.2 Forecasting with Plucking Models

Forecasts of yT +s based on a plucking model can be calculated by working with the

state space representation described in Chapter 2, in a similar fashion to the method previously described for linear models.

For plucking models, the state-space model consists of two equations an observa-tion equaobserva-tion and a state equaobserva-tion:

yt= hTξt (4.20)

ξ_t= µ_S_t+ Fξ_t₋₁+ vt (4.21)

vt∼ N(0, QSt) (4.22)

Solving the state equation (4.21) by recursive substitution yields: ξ_t+s = Fs−1µ_S_t+1+· · · + F1_µ

St+s−1+ µSt+s+ F

s_ξ

t+ Fs−1vt+1+· · · + F1vt+s−1+ vt+s.

(4.23) It follows, that the minimum MSE projection of ξt+son ξtconditional on ST +1, . . . , ST +s

is E(ξt+s|ξt, St+1, . . . , St+s) = Fs−1µSt+1 +· · · + F 1_µ St+s−1 + µSt+s+ F s_ξ t. (4.24)

Here the projection depends on ξ_t and the future states of the economy. From the law of iterated expectations, the optimal s-period ahead forecast conditional on ST +1, . . . , ST +s is ˆ ξT +s|T,ST +1,...,ST +s = F s−1_µ ST +1+· · · + F 1_µ S_{T +s−1} + µST +s+ F s_ˆ_ξST T|T, (4.25)

where ˆξS_TT_|T is an inference on ξT conditional on yT. The value of ˆξ ST

T|T is available

from the final iteration of Kim’s Filter applied to y1, . . . , yT.

To forecast the observed vector yT +s recall from the observation equation (4.20)

that

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The optimal s-period ahead forecast of yT +s conditional on ST +1, . . . , ST +s is then ˆ yT +s|T,ST +1,...,ST +s = h T_ˆ_ξ T +s|T,ST +1,...,ST +s (4.27) = hT_Fs−1_µ ST +1 +· · · + h T_F1_µ ST +s−1 + h T_µ ST +s+ h T_Fs_ˆ_ξST T|T. (4.28) This forecast is conditional on future states of the economy ST +1, . . . , ST +s. An

unconditional forecast can be achieved in theory by integrating out ST +1, . . . , ST +s

from equation (4.27). However, such integration is numerical intensive. Instead, a simulation technique can be used to generate an unconditional forecast. The simula-tion approach has an added advantage over numerical integrasimula-tion in that it provides a means to calculate prediction intervals.

The simulation procedure I used draws a sequence ST +1, . . . , ST +16 from the

first-order Markov-switching process:

P (St = 1|St−1 = 1) = p (4.29)

P (St= 0|St−1 = 0) = q. (4.30)

For s = 1, . . . , 16 the optimal s-period ahead forecasts ˆyT +s|T,ST +1,...,ST +s is evaluated

using ˆ yT +s|T,ST +1,...,ST +s = h T_Fs−1_µ ST +1+· · · + h T_F1_µ ST +s−1+ h T_µ ST +s+ h T_Fs_ˆ_ξST T|T. (4.31)

The procedure is repeated M times to obtain M realizations for each ˆy_{T +s|T,S}_{T +1},...,ST +s

denoted by _{ˆy_{T +s}(i) _|T,S

T +1,...,ST +s}

M

i=1. The point forecast ˆyT +s|T,ST +1,...,ST +s is then the

sample average of {ˆy_{T +s|T,S}(i) _{T +1},...,ST +s}

M

i=1. To calculate the 95% prediction interval

the simulation realizations are ranked and the lower bound is taken as the realization where 2.5% of the sample falls to the left and the upper bound is taken as the realization where 2.5% of the sample falls to the right. This procedure bypasses an extra complication by assuming that θ is known, which means that the uncertainty from the estimation of parameters is ignored.

Forecasts of yT +s for s = 1, . . . , 16 and prediction intervals based on the T most

recent values for the U.S., Canada and Australia are presented in Figure ??.

Figure 4.1 displays the 16 quarter ahead forecast of log GDP for the U.S.. The correlated innovations plucking model forecast diﬀers significantly from the symmetric

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model based forecast, predicting a slower recovery from the current recession. How to explain this slow recovery? The probability that the U.S. is in a normal state at time 2011:1 is 0.9340 and the probability of transitioning from a normal state to a recession state is .0308, so it is unlikely that the economy is being hit by negative shocks. This leaves the trend growth rate as a possible explanation. In 2011:1 the estimated trend growth term conditional on being in a normal state is 0.0027 comparing this with the average of the past 20 years, 0.0062, trend growth is seen to be at an all time low. The model indicates that the U.S. is experiencing a productivity slowdown.

Figure 4.2 displays the 16 quarter ahead forecast of log GDP for Canada. The plucking model forecast and the symmetric model based forecast are very similar. This linear trend growth is well explained by the high probability that Canada is in a normal state and the 2011:1 estimated trend growth term. The probability that Canada is in a normal state at time 2011:1 is .9999 and the probability of transitioning from a normal state to a recession state is .0163. The 2011:1 estimated trend growth term conditional on being in a normal state is 0.0045 close to the average of the past 20 years, 0.0059. Together these facts explain the steadily increasing forecast output over the 16 quarter forecast period. It is worth noting that unlike the symmetric model forecast, the plucking model upper and lower prediction interval limits are not symmetric around the forecast path. The forecast path is at the upper prediction interval limit over the entire forecast period. The fact that the trend ceiling hypothesis was not rejected for Canada explains this observed asymmetry. The forecast path can be viewed as a ceiling maximum level for output.

Figure 4.3 displays the 16 quarter ahead forecast of log GDP for Australia. The plucking model forecast predicts higher growth than the symmetric model based fore-cast. Australia has a much lower probability of being in a normal state at time 2011:1, .5335, than the U.S. or Canada. Estimates of trend growth for normal and recession states are .0095 and .0122, respectively. Both these estimates are higher than their re-spective averages of the past 20 years, 0.0086 and 0.0109, explaining the high growth forecast.

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Time log GDP 2006 2008 2010 2012 2014 9.4 9.5 9.6 9.7 9.8 Correlated Pucking Symmetric

Figure 4.1: Forecasting U.S. GDP: A series of 16 realizations from the corresponding first-order Markov-process St was simulated starting at time T + 1. The state ST was

randomly drawn from a Bernoulli distribution with probabilities P (ST = 0|ψT) and

P (ST = 1|ψT) taken from the corresponding Kim’s filter output. Subsequent states

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Time log GDP 2006 2008 2010 2012 2014 27.8 27.9 28.0 28.1 28.2 KN Pucking Symmetric

Figure 4.2: Forecasting Canadian GDP: A series of 16 realizations from the corre-sponding first-order Markov-process Stwas simulated starting at time T +1. The state

ST was randomly drawn from a Bernoulli distribution with probabilities P (ST = 0|ψT)

and P (ST = 1|ψT) taken from the corresponding Kim’s filter output. Subsequent

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Time log GDP 2006 2008 2010 2012 2014 12.4 12.5 12.6 12.7 12.8 12.9 13.0 KN Pucking Symmetric

Figure 4.3: Forecasting Australian GDP: A series of 16 realizations from the corre-sponding first-order Markov-process Stwas simulated starting at time T +1. The state

ST was randomly drawn from a Bernoulli distribution with probabilities P (ST = 0|ψT)

and P (ST = 1|ψT) taken from the corresponding Kim’s filter output. Subsequent

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4.2 Forecasting Inflation

Stock and Watson (2010) studied the state of inflation forecasting in the U.S.. They found that multivariate forecasting models which predict inflation using activity vari-ables did not on average improve upon a univariate benchmark model. The activity variables they considered included output gaps and unemployment gaps. In this sec-tion, a multivariate forecasting model using an activity variable derived from the correlated innovations plucking model is proposed. Using quarterly data the pseudo out-of-sample performance of the multivariate forecasting procedure with the new activity variable applied to forecasting U.S., Canadian and Australian inflation data is compared with a univariate IMA(1,1) forecasting model.

The three seasonally adjusted inflation data sets analyzed are: U.S. personal con-sumption expenditure deflator for core items PCE-core 1947:2 through 2011:1 (U.S. Department of Commerce: Bureau of Economic Analysis, Series ID: PCECTPI), Australia consumer price index 1969:4 through 2011:1 (Australian Bureau of Statis-tics:Series ID A2325846C), Canada GDP deflator 1961:2 through 2011:1 (Statistics Canada). All analysis uses quarterly inflation data. The quarterly rate of inflation at an annual rate is approximated by πt= 400 log(Pt/Pt−1) where Pt is the quarterly

price index.

4.2.1 Univariate Forecasting

The IMA(1,1) model appears frequently in the literature and has historically served as a forecasting benchmark. Stock and Watson (2007) selected the IMA(1,1) model for quarterly U.S. GDP price index inflation. They argued that the apparent unit root in πt and the negative first-order autocorrelations and generally small higher-order

autocorrelations of _�πt = πt− πt−1 suggest that the inflation process might be well

described by the IMA(1,1) process,

�πt= at− θat−1 (4.32)

where θ is positive and at is serially uncorrelated with mean zero and variance σa2.

This model can be expressed as a a linear Gaussian state-space model and forecasts of πt+1 made using data through time t, ˆπt+1|t, can be calculated from the Kalman

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4.2.2 Multivariate Forecasting

Multivariate forecasting models look to improve on univariate forecasts based only on past inflation by incorporating measures of economic activity into the basic model. Activity measures which appear frequently in the inflation forecasting literature in-clude: output gap and unemployment rate. These measures of economic activity are thought to be useful in forecasting inflation. For example, when the output gap is large or the unemployment rate is relatively low, prices in general often rise, leading to higher rates of inflation. Conversely, periods when output gap is negative or the unemployment rate is high tend to be disinflationary.

Stock and Watson (2010) considered a multivariate forecasting model in an activ-ity variable, xt, to predict the forecast errors from a univariate forecast of inflation.

Their model for forecasting inflation over the next quarter, πt+1, is

πt+1− τt|t = βxt+ �t+1 (4.33)

where �t+1 is an error term. At the end of the sample (date T ), the forecast of πT +1 is

computed directly using the estimated forecasting equation. The trend τt|t is derived

from the univariate IMA(1,1) model described in the preceding section.

The selection of a predictive activity gap is essential to successful forecasting. Stock and Watson (2010) used a gap measure which they refer to as a “recession gap” computed as the deviation of output from trend estimated using the Hodrick-Prescott filter. The Hodrick-Hodrick-Prescott filter is a linear filter and the trend derived from it is subject to the same criticism given to Clark’s model.

Following the example of Stock and Watson (2010) a multivariate model was constructed using the transitory component ct from the plucking models as a gap

measure. The unobserved components model with Markov switching does a better job of describing GDP than linear models and so multivariate forecasting using an output gap derived from this model is expected to outperform linear models.

4.2.3 Pseudo Out-of-Sample Forecasting

The forecasting performance of the multivariate forecasting model is examined rela-tive to the IMA(1,1) model, in pseudo out-of-sample forecast experiments on U.S., Canadian and Australian data as in Stock and Watson (2010). Pseudo out-of-sample forecast experiments simulate out of sample forecasting by performing all model

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spec-ification and estimation using data through date t, making a 1-step ahead forecast of πt+1 using the model, then moving forward to date t + 1 and repeating this through

the sample. The centred rolling root mean squared error (RMSE) statistic, a mea-sure of forecast error, is used to evaluate the performance of the forecasts. This is the square root of a weighted moving average of the squared pseudo out-of-sample forecast error: rollingRM SE(t) = � � � ��t+7 s=t−7 K(|s − t|/8)(πs+1− ˆπs+1|s)2/ t+7 � s=t−7 K(|s − t|/8), (4.34)

where K is the biweight kernel, K(x) = (15/16)(1_{− x}2₎2₁₍_{|x| ≤ 1) and ˆπ}

t+1|t is the

pseudo out-of-sample forecast of πt+1 made using data through date t. The higher

the RMSE, the higher the deviation between the forecast values and the actual values on average.

The results of the pseudo out-of-sample forecast experiments are displayed in Figures 4.4(a), 4.5(a) and 4.6(a). Figure 4.4(a) shows the U.S. inflation series 1947:2 through 2011:1 and the univariate and multivariate pseudo out-of-sample forecasts. The first forecast date for the multivariate forecast is 1962:1 the latest date necessary for the shortest regression to have 40 observations. For the U.S. both univariate and multivariate methods produce forecasts of inflation that follow relatively smooth paths. The precipitous drop in inflation in 2008:4 is the only event that produces forecasts that deviate significantly from the smooth path. Low inflation is forecast for 2009:1 lagging 1 quarter behind the observed minimum. Forecasts for Canadian and Australian data follow similarly smooth paths.

Forecast accuracy measured by the rolling RMSE is displayed in Figures 4.4(b), 4.5(b) and 4.6(b). Figure 4.4(b) shows rolling RMSEs from univariate and multivari-ate pseudo out-of-sample forecasts of U.S. inflation. The forecasting performance of both methods varies markedly over time. Several years ago, the rolling RMSEs were at historic lows, but they have recently crept up to levels of the early 1980s. The multivariate model improves upon the IMA(1,1) model during the disinflations of the early 1970s, the early 2000s and during the current recession. Figure 4.5(b) shows rolling RMSEs from forecasting Canadian inflation. The multivariate model improves upon the IMA(1,1) model during the disinflation of the early 1980s and 1990s. The multivariate model did relatively poorly during the late 1990s. Figure 4.6(b) shows rolling RMSEs from forecasting Australian inflation. The multivariate model does

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relatively poorly during the mid 1990s. This period corresponds to the period of unusually slow recovery following the recession of the early 1990s. The increase in inflation during the recession was atypical for this stage of the business cycle. For all three countries, the improvements of the multivariate forecasts is the greatest in recessions.

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Time Inflation, percent 1950 1960 1970 1980 1990 2000 2010 − 10 0 10 20 Actual Inflation Univariate Forecast Multivariate Forecast

(a) Inflation Forecasts

Time RMSE 1960 1970 1980 1990 2000 2010 1 2 3 4 5

Univariate Forecast RMSE Multivariate Forecast RMSE

(b) RMSE

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Time Inflation, percent 1960 1970 1980 1990 2000 2010 0 5 10 Actual Inflation Univariate Forecast Multivariate Forecast

Time RMSE 1980 1990 2000 2010 0.5 1.0 1.5 2.0 2.5

Univariate Forecast RMSE Multivariate Forecast RMSE

(b) RMSE

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Time Inflation, percent 1970 1980 1990 2000 2010 0 5 10 15 20 Actual Inflation Univariate Forecast Multivariate Forecast

Time RMSE 1985 1990 1995 2000 2005 2010 1 2 3 4

5 Univariate Forecast RMSE

Multivariate Forecast RMSE

(b) RMSE

Figure 4.6: 1 quarter ahead forecasts of Australian inflation and rolling RMSEs. The spike in inflation observed in the year 2000 is the result of Australia’s introduction of a goods and services tax. This spike in inflation results in poor forecasts of inflation.

Business cycle asymmetry: state-space models with Markov switching.

Contents

List of Tables

List of Figures

Introduction

1.1

Business Cycle Asymmetry

1.1.1

Formal Model

1.1.2

Correlated Innovations

1.2

Structure of the Thesis

Chapter 2

The Model

2.1

The Model

2.1.1

Extended Model

2.1.2

Identification

2.1.3

Estimation

Chapter 3

Experiments

3.1

Model Selection

3.2

Interpretation of Parameter Estimates

3.3

Permanent and Transitory Components

3.4

The Great Moderation

Chapter 4

Forecasting

4.1

Forecasting GDP

4.1.1

Forecasting with Linear Models

4.1.2

Forecasting with Plucking Models

4.2

Forecasting Inflation

4.2.1

Univariate Forecasting

4.2.2

Multivariate Forecasting

4.2.3

Pseudo Out-of-Sample Forecasting