Regional Business Cycles in Canada: A Regime-Switching VAR Approach

Transcription

1 JRAP 47(1): MCRSA. All rights reserved. Regional Business Cycles in Canada: A Regime-Switching VAR Approach Ronald H. Lange Laurentian University Canada Abstract: This study uses a Markov-switching methodology to capture the asymmetric nature of provincial business cycles in Canada. The estimations identify two- and three-regime provincial business cycles as well as some provincial economies that do not experience explicit cycle phases. Despite the asychronicity of provincial business cycles, concordance indices identify a very close cyclical pattern between most provinces and Canada as the reference economy, and maximum correlation coefficients indicate that recessions in Ontario, which has a relatively large concentration of manufacturing, lead overall recessions in Canada and in some of the other provinces. The findings in this study suggests that indicators of the business cycle in the most representative region could be a useful policy tool for forecasting aggregate economic activity. 1. Introduction In general, regional business cycles would be expected to mimic national cycles because of the general transmission of cyclical impulses, such as monetary policy or consumer and producer sentiment shocks, through the national economy to the regional level. However, disparities in regional business cycles in Canada could be attributed to either idiosyncratic shocks, such as the price of oil in the case of Alberta, or to differences in the industrial composition of the regions, such as the concentration of manufacturing in central Canada. The purpose of this study is to determine the extent to which provincial and regional economies in Canada move according to a common driving force and the extent to which there are different regional cycle patterns. The analysis of business-cycle synchronization could provide useful information for policy-makers in determining the regions most sensitive to economic shocks or policy changes and the regional business cycles that generally lead the national cycles. This study uses a Markov-switching methodology to capture the asymmetric nature of provincial business cycles in Canada. Regime-switching models have now become useful tools for analyzing the synchronization of national economies, as seen in Artis, Krolzig, and Toro ( 004), Artis, Marcellino, and Proietti (004), Camacho, Perez-Quiros, and Saiz (006), and Krolzig and Toro (005) for countries in the European Monetary Union (EMU), in Smith and Summers (005) for other industrialized countries, and for regional economies in the United States as analyzed by Owyang, Piger and Wall (005), and Hamilton and Owyang (01). The Markov-switching methodology has an advantage over classical business-cycle approaches that identify business cycles with coincident composite indicators for an

2 6 Lange economy 1 because it offers a joint statistical model for an economy. The methodology measures economic activity as function of a latent (unobserved), stochastic regime variable that indicates whether the economies are in a recessionary or an expansionary phase. This econometric approach can capture important, stylized business-cycle features from a group of economic time series, such as the regime probability, the synchronicity of the business cycles among different economies, the structural break of the business-cycle phases, the transition probabilities from recessions to expansions, the behaviour of volatility in the different business-cycle phases, and the duration of the cyclical phases in each economy. Overall, the regime-switching methodology is able to provide some useful stylized features of the business cycles of different economies. To preview the results, Markov-switching univariate estimations identify two- and three-regime provincial employment cycles. Also, some provincial economies are found to be unsynchronized with the national economy because they do not experience explicit employment-cycle phases. Despite the asychronicity of provincial employment cycles, concordance indices identify a very close cyclical pattern between most of the provinces and Canada as the reference economy, and maximum correlation coefficients indicate that recessions in Ontario lead overall recessions in Canada and in some of the provinces. The Markov-switching VAR models capture common regime shifts in the stochastic process of the employment growth for the two groups of provinces. The regimes are very persistent, with only small probabilities of shifting from one regime to another. The findings in this study suggest that indicators of the business cycles in the largest and most representative region, Ontario, may be a useful policy tool for forecasting national economic activity. However, heterogeneity in the employment cycle for some of the provinces suggests that policies focusing on smoothing provincial or regional fluctuations might be more suitable than general monetary or fiscal policies that focus only on the national business cycle. The following section briefly reviews some previous research on identifying national and regional business cycles. Section 3 outlines the Markovswitching technique for both univariate (AR) and multivariate (VAR) estimations. Section 4 discusses the data and the preliminary specification of the models. Section 5 presents the estimation results and the synchronization measures of the business-cycle phases for the univariate estimations and the estimation results for the multivariate estimations. The final section of the study briefly discusses some of the implications of the results for future policy and research.. Previous related research Since the seminal work by Hamilton (1989) on U.S. business cycles, numerous studies have applied Markov-switching models to the study of business cycles. Some useful examples include Goodwin (1993), Durland and McCurdy (1994), Filardo (1994), Kim and Nelson (1999), and McConnell and Perez-Quiros (000). Within this framework output growth is modelled as an auto-regressive process subject to parameter shifts, usually in the conditional mean and, sometimes, in the variance, as determined by a latent Markov process. The Markov-switching technique is particularly useful because it captures asymmetries in both the depth and duration of growth phases and fits the data better than linear models of output growth. In general, the literature on the nature of synchronization between business cycles using a bivariate or multivariate regime-switching framework has been quite limited. Phillips (1991) estimated a bivariate version of Hamilton s (1989) univariate regimeswitching model in which the unconditional means of real GDP growth for a pair of economies are driven by an unobserved, four-state Markov process consisting of a high-growth and low-growth state for each country. Phillips (1991) points out two extreme cases presented in the literature: the case of complete independence (two independent Markov processes are hidden in the bivariate specification) and the case of perfect synchronization (only one Markov process for both variables). The evidence of synchronization is assessed in Phillips by testing restrictions on the transition matrix, which would imply a common twostate Markov process affecting output growth in each country. Krolzig (001), Artis, Krolzig, and Toro ( 004), and Krolzig and Toro (005) extend Phillip s (1991) analysis to the multivariate case with common regime dynamics and assuming perfect synchronization of Markov states across countries in the EMU. A 1 See Burns and Mitchell (1946) for the original contribution and Diebold and Rudebush (1996) for a review of some of the literature on the classical approach.

3 Regional Business Cycles in Canada: A Regime-Switching VAR Approach 63 multivariate Markov-switching model was also applied to countries in the EMU by Camacho, Perez- Quiros, and Saiz (006) and to various industrialized countries by Smith and Summers (005). Two recent studies have also focused on economic activity synchronization when the business cycle is disaggregated at the regional level. Owyang, Piger, and Wall (005) investigate the evolution of the individual business-cycle phases of U.S. states using growth rates of monthly coincident indices. By following a univariate Markov-switching approach, the authors find that U.S. states differ significantly in the timing of switches between expansions and recessions and also differ in the extent to which phases in state business cycles are sychronous with those of the national economy. Using a multivariate approach, Hamilton and Owyang (01) study the propagation of state recessions in the U.S. using the quarter-toquarter growth rates of non-farm payroll employment. They find that differences across states appear to be a matter of timing and can be grouped into three clusters, with some entering recession or recovering before others. These studies generally rely on two measures of synchronization, as proposed by Harding and Pagan (006): correlation (the extent to which turning points in the two series occur near each other) and concordance (the fraction of time that two series are in the same state). For example, Guha and Banerji (1998, 1999) estimate different two-state Markov-switching univariate models on non-farm employment data for some states in the United States and another twostate model for the aggregate ( national ) data. They then use a bootstrap procedure to compare the residuals from the Markov-switching regional equations of employment data to those from of a national equation and compute cross-correlations between the probabilities of being in a recession as a measure of synchronization. They find that the business-cycle phases of California, New York, Illinois, and Florida have been different than those of the United States as a whole. However, while some idiosyncrasies exist, they find that regional business cycles in the United States for the most part bear a reasonable resemblance to the national cycle as identified by the NBER using aggregate data. Tests of synchronization using correlation coefficients for cycles are also used on industrial production for various industrialized countries by Harding and Pagan (006) and on rate of growth of industrial production for the Euro countries by Artis, Krolzig, and Toro ( 004). Harding and Pagan (006) also propose a concordance test for the hypothesis that cycles are either unsynchronized, experiencing independent business cycle phases, or perfectly synchronized. Their concordance measure corresponds to the probability that the phases of regional business cycles and national business cycles are the same; that is, they are ergodically synchronized. The index is applied by Harding and Pagan (006) to industrial production for various countries, by Artis, Krolzig, and Toro ( 004) to the growth rate of monthly industrial production, by Artis, Marcellino, and Proietti (004) to quarterly GDP for the euro area, by Krolzig, and Toro (005) to the growth rate of industrial production in nine European countries, by Smith and Summers (005) to GDP for selected industrialized countries, and by Owyang, Piger, and Wall (005) to the growth rate of monthly coincident indices for U.S. states. The results from the studies on regional economies generally find the existence of interdependent cycles in the United States, where state-level business-cycle experiences are similar to those of the nation. The most idiosyncratic recession experiences amount to differentials in timing around the national recessions, where some states enter recessions before the rest of the nation. The studies also find a cluster of states, characterized by an important role for oil production in their economies, that enter and exit recessions independently from the nation. 3. Empirical methodology The first step in the empirical strategy is to estimate univariate, Markov-switching auto-regressions (MS-AR) for each of the provinces to give a statistical characterization of the business- or economic-cycle phases. The univariate characteristics can identify common regime shifts in the stochastic process of interdependent regimes. The second step involves estimating multivariate models (MS-VAR) that capture the common regime shifts for groups of provinces from the univariate estimations. The m-regime, p-th order Markov-switching autoregression for the estimations is the following intercept-switching, heteroskedastic auto-regression, or MSIH(m)-AR(p) in the terminology of Krolzig (1997): p y t = ν + i=1 A i y t i + ε t, (1) where y t is an observed time series [y 1, y, y T ], [ν 1 (s t ), ν (s t ),, ν T (s t )] is a column of regime-dependent intercepts, the A t s are columns of auto-

4 64 Lange regressive parameters, and ε t = [ε 1t, ε t,, ε T ] is a vector of Gaussian white-noise processes ε~nid(0, σ (s t )) conditioned on s t, which is an unobserved discrete-regime variable evolving according to a time-varying, state-dependent process. Regimedependent heteroskedasticity is considered, although there are not relevant differences of volatility between regimes for some provinces. The contractions and expansions are modelled as a function of a latent or unobserved state variable s t that indicates switching regimes in the stochastic process generating the growth of the economic activity index of the economy. In the m-regime, p-th order Markov-switching vector auto-regression for the multivariate estimations, equation (1) becomes a matrix equation for an intercept-switching, heteroscedastic, vector auto-regression, or MSIH(m)-VAR(p), where y t = [y 1t, y t,, y Kt ] is a k-dimensional vector time series for t = 1, T, ν(s t ) = [ν 1 (s t ), ν (s t ),, ν K (s t )] is a k- dimensional column for regime-dependent intercept terms and s t is an unobserved discrete-regime variable evolving according to a time-varying, state-dependent process, A i is a k x k matrix of auto-regressive parameters, and ε t = [ε 1t, ε t,, ε Kt ] is a k- dimensional vector of Gaussian white noise processes for each regime i with covariance matrix Σ conditioned on s t, ε~nid(0, Σ(s t )). In this specification of the MSIH-VAR model, the autoregressive parameters are assumed to be regime independent. For m regimes, s t ε {1,, m}, the matrix equation for equation (1) may be summarized as y 1t ν 1 + j A 1j y t j + y t = [ ] = [ y p mt ν m + A 1j y t j + p j 1/ 1 ε t 1/ m ε t ], () where the intercepts ν i for i = 1,, m in equation () are simply the regime-weighted average of the means of the residuals from the VAR. Unlike regime-invariant models, the MSIH-VAR allows for shocks to the white-noise series ε t. The specification has the advantage of allowing the mean intercept to smoothly approach a new level after the transition from one state to another since a once-and-for-all regime shift is identical to an equivalent shock in the white noise series ε t. The description of the data-generation process is not completed by the observational equation (1). A In contrast, a mean-switching, heteroskedastic, vector auto-regressive model has a multiplicative relationship between the VAR coefficients and the intercepts that allows for an immediate onemodel for the regime-generating process is needed to allow for inference about the evolution of regimes from the data. The special characteristic of the Markov-switching model is the assumption that the unobservable realization of regime s t ϵ {1,, m} is governed by a discrete-time, discrete-state, Markov stochastic process. Formally, the stochastic process is defined by the transition probabilities p ij = Pr (s t+1 = j s t = i), m j=1 p ij = 1 i, jϵ{1,, m} (3) More specifically, s t is assumed to follow an ergodic and irreducible m-state Markov-chain of order one with the transition matrix p 11 p 1 p 1m P = [ p 1 p p m ], (4) p m1 p m p mm where p im = 1 p i1 p i,m 1 for i = 1,, m. By inferring the probabilities of the unobserved regimes conditional on an available data set, it is then possible to reconstruct the regimes. For an ergodic Markov chain, regime shifts are persistent if p ij p ii for i j and not permanent if p ii 1 for all i. The two components of the MSIH(m)-VAR(p) model, the Gaussian multivariate model (1) as the conditional data generating process and the Markov chain (3) as the regime generating process, are estimated using a likelihood-based statistical method. The maximization of the likelihood function of the MS-VAR entails an iterative technique to obtain estimates of the intercepts ν(s t ) and variance-covariance matrices Σ(s t ), and the transition probabilities p ij governing the Markov chain of the unobserved states. The maximum-likelihood estimation (ML) is based on the implementation of the Expectation Maximization (EM) algorithm proposed by Hamilton (1990, 1994) for this class of models. Each iteration of the EM algorithm involves two steps. The expectation step involves a pass through the smoothing algorithm, using the estimated parameter vector for the VAR of the last maximization step in place of the unknown true parameter vector to infer the hidden Markov chain. In the maximization step, an estimate of the parameter vector is derived as a solution of the firstorder conditions of the likelihood function, where the conditional regime probabilities are replaced with the time, permanent jump in the process mean after a change in regime.

5 Regional Business Cycles in Canada: A Regime-Switching VAR Approach 65 smoothed probabilities derived in the last expectation step. These two steps are repeated until convergence is achieved for the maximum-likelihood function. 4. Data and preliminary specification The data used to measure provincial-level business cycles are the seasonally adjusted total employment for both sexes, 15 years and over. 3 The estimation period is from 1976:5 to 010:6 for a total sample of 410 observations. The series are the log differences multiplied by 100 so that they can be interpreted as month-to-month growth rates. Table 1 presents an overview of the four sectors comprising about 40 per cent of the level of employ- ment in the Canadian provinces in 010. The main feature is the relatively large percentage of employment in manufacturing in central Canada (Ontario and Quebec) relative to the other provinces, close to the level of employment in the retail industries in those provinces. Employment in retail and government is by far the largest percentage of employment in all provinces. With the exception of Saskatchewan and Prince Edward Island, employment in agriculture, fishing and forestry is a very small percentage of total employment. Employment in the oil and gas industries in Alberta is about.75 per cent of total employment (not shown), the largest proportion among the provinces. Table 1. Composition of the level of selected employment in Canada for 010. Region Total Manufacturing Retail Government Agriculture and % % % Fishing % Canada 17,371, Ontario 6,735, Quebec 3,88, Nova Scotia 465, New Brunswick 360, Prince Edward Island 69, Newfoundland 1, Manitoba 650, Saskatchewan 547, Alberta,086, BC,9, Notes: Percentages are of the total employment in each region. An issue of paramount difficulty in specifying Markov-switching models is the choice of the number of regimes and lags. The likelihood-ratio test statistic for testing the number of regimes does not possess an asymptotic χ distribution because of the existence of a nuisance parameter under the null hypothesis. This study uses the general-to-specific approach to econometric modelling proposed by Krolzig (1997). A univariate ARMA analysis was performed on each of the employment time series. The test procedure relies on the following theorem: in the class of MSIH-AR models, there exists for any ARMA(p *, q * ) representation 3 The measure most synonymous with GDP at the provincial level is gross provincial domestic product. Unfortunately, the data is available only at an annual frequency in Canada, making it nonviable for a study of business cycles. Hamilton and Owyang (01) also use seasonally adjusted quarter-to-quarter growth rates of payroll employment to measure state-level business cycles in the U.S. with p q 1 a unique MSIH(m)-AR(p) with m = q * +1 and p = p * -q *. The Akaike information (AIC) and the Schwartz criteria (SC) are employed to assist in choosing the appropriate order of the ARMA(p, q) processes, which enables selection of the MSIH-AR models that can be expected to be consistent with the data. 4 Table presents the ARMA estimations for the provinces that could be captured with either two- or three-regime MSI(m)-AR(p) estimations. 5 In general, the results from the ARMA estimations are quite mixed, with the information criteria suggesting two 4 See Krolzig (1997, p. 130, for the theorem). 5 Newfoundland, Prince Edward Island, New Brunswick, and Saskatchewan did not exhibit explicit employment cycles over the sample period. See Table 1A in the Appendix for the two-regime estimates for those provinces, where estimates for at least one growth regime is insignificant.

6 66 Lange regimes for most provinces. However, the AIC criterion indicates three regimes for Quebec and British Columbia, while both the AIC and SC suggest three regimes for Manitoba. Since the general-to-specific approach does not provide consistent guidance to the choice of the number of regimes and lags, the final specification of the univariate models was based on the significance of the intercepts, the variances of the Gaussian innovations, and the order of the auto-regessions, as well as the value of the likelihood function, for the estimations of the MSIH(m)-AR(p) for each province. Table. ARMA representation pre-selection for univariate specification. Ontario Quebec N. S. Manitoba Alberta B.C. ARMA(p, q) SIC AIC SIC AIC SIC AIC SIC AIC SIC AIC SIC AIC MSI(M)-AR(p) ARMA(1,1) MSI()-AR(0) ARMA(,1) MSI()-AR(1) ARMA(3,1) MSI()-AR() ARMA(4,1) MSI()-AR(3) ARMA(5,1) MSI()-AR(4) ARMA(5,) MSI(3)-AR(3) ARMA(6, ) MSI(3)-AR(4) Notes: SIC is Schwartz information criterion, AIC is Akaike information criterion, and bold is for lowest values. 5. Empirical results 5.1. Markov-switching univariate estimations The approach in the estimations is to search for the best model in terms of the number of regimes and lag lengths to capture the employment-cycle features for Canada as a whole and the provinces. The main conclusion is that a two-regime model with contractions and expansions is not adequate to capture the employment cycle for Canada as well as for some provinces. Therefore, the model was extended to three regimes, one for a recession cycle and two cycles corresponding to moderate- and high-employment growth regimes. Figures 1 and plot the smoothed probabilities for the three- and two-regime estimations, respectively. The smoothed probabilities in the top panel of Figure 1 indicate three recession regimes: the early 1980s, which was triggered by the second major oil-price shock; the early 1990s, which was triggered mainly by the aggressive monetary policy needed to achieve the adoption of explicit inflation targets in 1991; and which was triggered by the financial crisis in the United States. The high-employment growth regime 3 in the bottom panel occurred mainly in the late 1970s and early 1980s around the first recession and again following the latest recession. Figure 1A in the Appendix indicates that the employment growth regimes closely correspond to the business cycles regimes for Canada from a univariate estimation with quarterly real GDP growth. Although the provincial-level recessions tend to be associated with national recessions, there is still some province-specific variation in the timing and length of recessions. Specifically, individual provinces can switch into or out of recession long before or after the nation as a whole does, or a province can be in a high growth regime when the rest of the country is only experiencing moderate employment growth. Since Ontario and Alberta are quite diverse economies, at times they did not closely follow the national economy. For example, Alberta experienced an energy-related period of high growth in that was not experienced by the other provinces and the country. Also, some provinces experienced significant business cycle idiosyncrasies. As indicated in Figure, Quebec did not experience employment recessions in the early 1990s and in as the rest of Canada, while Manitoba and Nova Scotia only experienced the employment recession in the early 1990s.

7 Regional Business Cycles in Canada: A Regime-Switching VAR Approach 67 Regime 1 and Recessions Regime and Moderate Employment Growth Regime 3 and High Employment Growth Canada 1.0 Canada 1.0 Canada Ontario Ontario 1.0 Ontario Alberta Alberta 1.0 Alberta Figure 1. Three-regime provinces. Table 3 presents the results for the Markovswitching univariate estimations for Canada as whole and the six provinces. For Canada, the normal employment growth regime has an intercept mean of 9 per cent growth, the high-growth regime has a value of 0.14 percent and the highest variance, and the recession regime has an average growth of - per cent and about the same variance as the moderate growth regime. The employment cycles for Ontario in the second column closely mirror those for Canada, although the intercept in the recession regime is slightly more negative and the intercepts in the growth regimes are larger. Similarly, the variances in all three regimes are much larger than those for Canada. The three-regime estimations for Alberta indicate much higher mean growth regimes than either Ontario or Canada along with much larger variances for all three regimes. Overall, the three-regime estimations reveal much larger intercepts and variances for the individual provinces than those for Canada in aggregate, as expected. The two-regime models for Quebec, Nova Scotia, Manitoba, and British Columbia were sufficient to identify both recession and growth regimes. However, the negative intercept for the recession regime in Manitoba is not significant. Nevertheless, the negative recession intercepts for Quebec, Nova Scotia, and British Columbia are significant and are much larger than those for the three-regime estimations. The recession variance for Manitoba is at least twice the size of those in the other provinces, which is consistent with the prolonged period of negative growth in the late 1980s and early 1990s, shown in Figure. Quebec experienced only the first recession and British Columbia only the first and last recession periods.

8 68 Lange Quebec Quebec British Columbia British Columbia Manitoba Manitoba Nova Scotia Nova Scotia Figure. Two-regime provinces. Table 3. Markov-switching estimates for MSIH(3)- and MSIH() AR(p). ν 1-4 (-3.74) ν 90 (6.14) ν (6.99) σ 1 33 (3.3) σ 3 (8.51) σ 3 43 (7.67) ρ 0.376(3) () Canada Ontario Alberta Quebec N.S. Manitoba B.C (-4.73) (8.98) 5 (8.47) (4.09) 67 (8.80) 0.10 (7.68) (0) -0.1 (-.6) (9.38) (9.68) 0.9 (5.40) (9.77) 0.60 (6.34) (3) () -0.7 (-5.9) 0.14 (6.67) (.3) (13.87) -0.97(1) (9) (-4.66) 0.16 (4.66) (4.04) 00 (14.71) -0.34(1) () -59 (-0.51) 0.9 (4.4) 06 (3.39) 0.47 (11.47) -0.61(1) () (-.34) 0.98 (6.4) 0.63 (.75) 0.91 (10.93) -0.3(1) () LLF RCM Notes: ν i is the mean intercept for regime i, σ i is the variance for regime i, ρ is the sum of autoregressive coefficients, LLF is the log of the likelihood function, and RCM is the regime classification measure. Figures in parentheses for the intercepts and variances are t-statistics based on standard errors of the maximum function value and for the autoregressive coefficients are p-values for a Chi-square test on the sum of the coefficients.

9 Regional Business Cycles in Canada: A Regime-Switching VAR Approach 69 Figures 1 and indicate that the regimes are very well classified, with probabilities being relatively close to zero or one for both two- and three-regime estimations. The quality of the regime classification may be confirmed by the following the regime classification measure (RCM) proposed by Ang and Bekaert (00) for m states: T m i=1 ) RCM = 100m 1 ( p T t=1 i,t, (5) where p i,t = p(s t = i Y T ) is the unconditional probability of being in regime i. 6 The constants 900 for three regimes and 400 for two regimes serve to normalize the statistic to be between 0 and 100. A good regime classification is associated with low RCM values, with zero being a perfect classification and 100 implying that no information is revealed about the regimes. The RCMs for the univariate regime-switching estimations presented in the last row of Table 3 for Canada, Ontario, Quebec, and Alberta are virtually equal to zero, indicating very good regime classifications. The still relatively low RCMs for Nova Scotia, Manitoba, and B.C. are much higher, indicating that the regimes are not as well classified as those for the other provinces. 5.. Regime characteristics Table 4 reports for the provinces the measures of the persistence of the regimes: the expected number of months a regime prevails (duration) and the unconditional (ergodic) probability of being in the regime. The first three rows present the properties from the three-regime estimations for Canada, Ontario, and Alberta, and the last four rows present them for the two-regime estimations for the other provinces. The number of observations in the recession regime 1 is the largest for Alberta, accounting for over 0 per cent of the ergodic probability of a recession. The relatively large number of observations for Alberta probably reflects special factors, such as its dependence on oil and gas. The expected duration, calculated as 1/(1 p ii ), of the moderate and high growth regimes in Ontario are relatively long, at about 11.6 and 4 years, respectively, and quite similar in length to that for Canada as whole. As indicated in Figure, Quebec has had only one recession regime, with only 1 observations and a three percent ergodic probability, and the longest growth regime with a duration of 309 months or almost 6 years. In all provinces, the expected duration of an expansion is much longer than the expected duration of a recession. This indicates that although each province experiences relatively short recessionary periods, the baseline regime in each province is an expansion. The measures of regime characteristics in Table 4 indicate substantial differences among the provinces in both the duration and probability of being in the recession regime 1. Some recent research on both the impact of and recovery from the recession on U.S. states suggests some factors that may explain these relatively large differences. Connaughton (01), for example, found that states where a large percent of the population was employed in the manufacturing sector suffered the largest job losses as a result of the Great Recession. Similarly, Walden (01) found that states with a relatively large share of state GDP in manufacturing experienced the greatest labor market deterioration or the largest unemployment rate, which was consistent with studies on previous recessions. However, Walden (014) found that the states with concentrations in financial services and durable manufacturing were associated with the best recovery and greater employment growth after the Great Recession. Table 4. Regime properties. Number of Observations Ergodic Probabilities Duration Regime Canada Ontario Alberta Quebec N. S Manitoba BC Notes: The number of observations is observations in regime i with p ii > 0.5, the (unconditional) ergodic probability is the number of regime observations/the total sample number of observations, and duration is the expected duration of regime i (1/1 p ii ) based on estimates using the EM algorithm. 6 The RCM statistic is essentially a sample estimate of its variance.

10 70 Lange Although there appears to be some asynchronicity in the provincial employment cycles in Canada, it is still possible that these cycles do exhibit a certain amount of synchronization that can be interpreted as a sign that the provincial economies move together to some degree. In order to study the possible relationship between cyclical patterns of provincial employment activity, two measures of synchronization are constructed: the test statistic of concordance proposed by Hardy and Pagan (006) and correlation coefficients. The concordance index measures the proportion of time that two economies are in the same regime; that is, the degree of concordance of the regime shifts in employment growth. 7 Since the regimes are very well classified with RCMs close to zero in Table 3, a binary random variable S it is created for the smoothed probabilities of each time series for regimes in each province and the univariate ARs by assigning the value 1 if the smoothed probability P ii is greater than 50 per cent. 8 More formally, the rule assigns observations to regime i if Pr (s t = i Y T ) > 0.5. The concordance index for three regimes s t ε{1,, 3} is defined as follows: I 13 = T 1 T [ t=1((s 1t S t ) + (S 1t S 3t ) + (1 S 1t S t )(1 S 1t S 3t ))], (6) where T is the sample size and I 13 measures the proportion of time that the three regimes are in the same phase with probabilities greater than 50 per cent. For regions with only two regimes, that is, s t ε{1, }, the concordance is defined as: I 1 = T 1 T [ t=1 ((S 1t S t ) + (1 S 1t )(1 S t ))] (7) The indices correspond to the ergodic probability that the phases of provincial employment cycles and the national employment cycle, as well as pairs of provincial cycles, are the same. These indices capture similarities in the length of the cycles that are independent of the timing of the regime phases. Table 5 presents the concordance pattern for the estimations based on the univariate MS models that capture three regimes and those that identify only two regimes for the provinces. The most striking result is the high degree of synchronization of employment growth between Canada and Ontario captured by the three-regime index at 0.90 (in bold). This confirms the conventional view that employment growth in Canada is largely driven by employment growth in manufacturing in Ontario. However, the corresponding ergodic synchronizations from the two-regime indices also indicate very high degrees of concordance of the employment cycles among the other provinces and Canada, with indices ranging from 0.97 for Quebec to 1 for Alberta. The concordance patterns between pairs of provinces are also quite high, ranging from 0.97 for Quebec and British Columbia to 0.79 for Manitoba and Alberta. Table 5. Concordance indices for regime probabilities. Canada Ontario Quebec N. S. Manitoba Alberta Canada Ontario Quebec Nova Scotia Manitoba Alberta British Columbia Notes: The two-regime concordance indices are presented in the bottom triangle of the table and the three-regime indices are presented in bold in the upper portion of the table. 7 Harding and Pagan (004) and Krolzig and Toro (005) use concordance indices to capture the degree of synchronization of regional business cycles. 8 The arbitrary assumption that a province is in recession when its recession probability exceeds 0.5 is also adopted by Owyang, Piger, and Wall (005, footnote 11) and Smith and Summers (005).

11 Regional Business Cycles in Canada: A Regime-Switching VAR Approach 71 Although the concordance index is very easy to interpret and provides a first picture of synchronicity of provincial employment cycles, the correlation coefficient as a second measure of sychronization has the advantage of providing a statistical way of establishing whether the co-movements of the employment cycles are significant or not. Table 6 presents the correlations coefficients for the smoothed probabilities of being in the recession regime 1 between each pair of provinces and the p-values of Q-tests for the null hypothesis of no correlation. The first column presents the maximum correlation coefficients between Canada as the reference region and the provinces. The coefficients indicate that overall employment growth in Canada in the recession regime is preceded by the monthly declines in employment in Ontario and British Columbia. The maximum correlations also indicate that employment declines in Alberta lag those in Canada by as much as three months, while the declines in Nova Scotia precede those in Canada by several months, suggesting some idiosyncratic source for the decline in Nova Scotia for the early 1990s in Figure. The contemporaneous correlations in the second column indicate a relatively high correlation of about 0.5 between the employment declines in the recession regime in Ontario with those in the other provinces. The contemporaneous correlations in the third column also suggest a relatively high correlation been the recession regimes in Quebec and British Columbia. The pairwise correlations are typically smaller than those obtained with the concordance index, which are over 0.9, suggesting that the stronger correlation between employment cycles detected with the concordance index is biased by the value of the mean probabilities. Table 6. Correlation coefficients of being in the recession regime 1. Canada (lag/lead) Ontario (-1) () Quebec (0) () Nova Scotia (-5) () Manitoba (0) () Alberta 69 (3) () British 0.60 (-1) Columbia () Ontario Quebec N.S Manitoba Alberta 88 () 51 () 0.61 () () () () () () () 11 () 87 () 0.49 () () Notes: Terms in parentheses in the first column are maximum correlations at the lag (+) or lead (-). Terms below the correlation coefficients are p-values for the chi-squares of Q-tests of the null hypothesis of no correlation Markov-switching multivariate estimations The synchronicity of the regime shifts in the employment growth process of most of the provinces in Tables 5 and 6 suggests a system approach to the investigation of the common cycle of these provinces, which may constitute a national employment cycle. Consequently, the univariate model is generalized to three- and two-regime, Markov-switching, vector auto-regression models (MS-VAR) that may characterize the national business cycles as common regime shifts in the stochastic process of employment growth of interdependent provinces. Table 7 presents the three-regime multivariate estimations for Ontario and Alberta, the two-regime estimations for Quebec, Nova Scotia, Manitoba, and British Columbia, and, for comparative purposes, the univariate three-regime estimations for Canada as a whole from Table 3. The bivariate estimations for Ontario and Alberta are quite similar to the univariate estimations in Table 3, capturing recession employment regimes and two employment growth regimes for each province. However, the negative growth regime for Ontario is no longer significant at the 5 per cent level, and the two employment growth regimes are not in ascending order (that is, moderate and high growth), although the two intercepts are not significantly different from each other. On the other hand, the three-

12 7 Lange regime estimate for Alberta is in ascending order and is virtually identical to that reported for the univariate estimation in Table. It is noteworthy that the growth in regime 3 for Alberta is 78 per cent, over twice as high as its normal growth, which reflects an explosive expansion due to the special role that oil and gas plays in its economy. The smoothed probabilities from the three-regime and two-regime estimations are presented in Figures 3 and 4, respectively. The RCM of 0.1 for the regime probabilities in Figure 3 indicates that the three regimes for the bivariate estimation are almost perfectly classified and correspond very closely to those in Figure 1. Table 7. Markov-switching estimates for MSIH(3)- and MISH() -VAR(p). Three-regime VAR Two-regime VAR Canada Ontario Alberta Quebec N.S Manitoba B.C ν 1-4 (-3.74) -5 (-0.1) (-.5) -07 (-3.83) (-1.67) (-1.67) -97 (-3.65) ν 90 (6.14) (4.86) (9.69) (6.55) (4.0) (4.35) 0.79 (9.18) ν (6.99) 87 (.01) (11.39) σ 1 33 (3.3) (6.84) 0.45 (6.60) 0.7 (3.65) 0.89 (3.39) (3.15) 0.85 (3.33) σ 3 (8.51) 66 (8.98) (10.10) (1.67) (1.75) 8 (1.06) 0.90 (1.76) σ 3 43 (7.67) (7.68) 1 (7.10) ρ 0.376(3) () 0.88(3) () (3) () (1) () -0.8(1) () -0.70(1) () -0.05(1) () LLF RCM Notes: ν i is the mean intercept for regime i, σ i is the variance for regime i, ρ is the sum of autoregressive coefficients, LLF is the log of the likelihood function, and RCM is the regime classification measure. Figures in parentheses for the intercepts and variances are t-statistics based on standard errors of the maximum function value and for the autoregressive coefficients are p-values for a Chi-square test on the sum of the coefficients.

13 Regional Business Cycles in Canada: A Regime-Switching VAR Approach 73 Regime 1 and Recession Employment Growth Regime and Moderate Employment Growth Regime 3 and High Employment Growth Figure 3. Three-regime probabilities for the VAR estimation of Ontario and Alberta. The multivariate estimations for Quebec, Nova Scotia, Manitoba, and British Columbia in Table 7 are very close to the univariate estimations in Table 3, capturing both recession and growth employment regimes with virtually all of the intercept and variances Recession Growth being statistically significant. The RCM of 3.40 indicates that the regime probabilities in Figure 4 are also very well classified and capture the recession periods in the early 1980s, the early 1990s, and in Employment Growth Figure 4: Two-regime probabilities for Quebec, Nova Scotia, Manitoba and British Columbia

14 74 Lange Table 8 presents the transition matrices from Equation 4 and the regime properties for the threeand two-regime models for the provinces. The transition matrix for the three-regime estimation indicates that the regimes are very persistent with probabilities of at least 95 percent, with the recession growth regime being reached from the moderate growth regime with a small probability of five per cent. The moderate growth regime can be reached from both recession and high-growth periods with very small probabilities, while the high growth period can be reached from both the recession and moderate growth regimes with probabilities of 15 per cent. In both cases the transition matrix for the tworegime VAR also indicates that the recession and expansion periods are also very persistent, with the probabilities of a shift from one regime to the next being relatively small. The duration of a recession period for the threeregime estimation is about 1 months, while the average duration for the two growth periods are about the same at 3 and 35 months, respectively. Thus, the probability of Ontario and Alberta being in recession at the same time is about 5 per cent, while the probability of being in a moderate growth period is much higher (60 per cent) and the probability of a high growth period is about 15 per cent. For the two-regime group of provinces, the duration of a recession is much smaller at about eight months, as shown in Figure 4, with the ergodic probability of being in a recession of less than one per cent. However, the probability of being in an expansion period is about 93 per cent, and the duration of an expansion is much longer at about 100 months (8.3 years). Table 8. Transition matrices and regime properties. Regime Pi1 Pi Pi3 Observations Ergodic Probabilities three-regime VAR estimation Duration two-regime VAR estimation Notes: The number of observations is observations in regime i with p ii > 0.5, the (unconditional) ergodic probability is the number of regime observations/the total sample number of observations, and the expected duration of regime i (1/1 p ii ) in months is based on estimates using the EM algorithm. 6. Concluding remarks In this study, the approach innovated by Hamilton(1989) in his analysis of the U.S. business cycle was used to identify employment cycles in the Canadian provinces. The approach first consisted of fitting a Markov-switching regime process to the univariate data series of the provincial economies in order to capture the stylized facts about regime lengths, probabilities of switching, and the mean and volatility of employment growth in each province. The estimations identified two groups of provinces corresponding to two- and three-regimes of employment growth, with two expansions in the three-regime group corresponding to moderate and high growth. The regime probabilities are very well classified for both groups and capture the recession periods in the early 1980s due to the oil-price shocks, in the early 1990s when monetary policy was aggressively reducing inflation to achieve the explicit inflation targets that were adopted in 1991, and in due to the global recession triggered by the financial crisis in the United States. The identification of smoothed probabilities enabled the calculation of concordance indices and cross-correlations of those probabilities that indicate considerable synchronicity among the provincial employment cycles.

15 Regional Business Cycles in Canada: A Regime-Switching VAR Approach 75 In response to the finding of important synchronization in the employment cycles, the procedure was extended to fit Markov-switching VAR models to the data with the individual provincial series making up the VAR models. The objective in the estimations is to capture the extent to which there are common regime shifts in the stochastic process of the employment growth in each of the two groups of provinces. The VAR estimations allow for the analysis of the effect of the aggregation of provincial employment cycles into a common cycle that may be interpreted as a national cycle. The estimations for both three- and two-regime VAR models indicate that the regimes are very persistent with probabilities of remaining in a regime of at least 95 percent. The duration of a recession period is about two and one-half years for the three-regime group consisting of Ontario and Alberta, with the probability of both being in a recession at the same time of about 5 per cent. The average duration for the two growth periods is almost three years. The findings of this study have at least four interesting implications for economic behaviour and policy. First, there is the possibility of forecasting the national cycle on the basis of provincial or regional analysis. Once the regional cycles are identified, the largest and most representative province, such as Ontario, might be used to forecast aggregate activity since its business cycle appears to lead the national cycle. Secondly, the identification of regime probabilities for the turning points in the business cycle may have relatively large benefits for the business sector and policy makers. For example, since business people want to have the best assessments of current and future economic activity, even small forecast improvements may lead to large differences in profits because of recognizing the probability of regime switching. Similarly, recognizing the potential for regime switching may lead to better policy choices, such as providing greater access to employment insurance during economic downturns. Thirdly, the regime switching itself may affect the optimal rules for consumption and investment. For example, inventory investment may switch with regimes, as may agents ability to borrow. Finally, there appears to be some heterogeneity in the employment cycles among some of the provinces. Different parts of the country have manifested the signs of a downturn while others have not. Manitoba, for example, only experienced a prolonged downturn in the late 1980s and early 1990s, while most of the other provinces also experience recessions in the early 1990s and again in Analogously, some provinces have experienced an explosive expansion, while other provinces appear to have experience only a normal expansion. Alberta, for example, experienced a period of very high employment growth in due to the effect of higher resource prices on the oil and gas sector. These types of anomalies in the provincial employment cycles are relevant for economic policy because the usual response to an economic slowdown is for the monetary authority to loosen monetary policy to smooth out the national business cycle. However, a nationwide policy may not be needed if the focus is only on a particular region or province. This is demonstrated by the studies of regions in the U.S. by Carlino and DeFina (1998, 1999), Fratantoni and Schuh (003), and Owyang and Wall (004). They show that the impact of monetary policy can depend on the mix of states in and out of recession at the time the policy is implemented. Their results show that the impact can depend on the mix and size of firms and on the structure of banking in the states. A similar argument can be applied to the use of fiscal policy to smooth aggregate business cycles. The technical recession in the first half of 015 in Canada is a good example of this policy dilemma. The recession was triggered by the effects on Alberta of sharp declines in global oil prices, which suggests that an employment policy in Canada directed at the oil and gas sector and at Alberta in general might be more appropriate than either a general monetary or fiscal response. As a first pass to the study of regional business cycles in Canada, the methodology in this study has generally relied on the model for regime-switching intercepts and variances outline in Krolzig (1997). However, further empirical research should determine whether there is an equilibrium relationship between provincial and national employment levels. A Markov-switching vector error-correction model (MS-VECM) with a co-integrating vector for the logs of provincial and national employment levels along with a drift term for possible changes in productivity and an error correction mechanism would be able to address this issue. In the MS-VECM framework, regime disequilibria for the provinces would also be adjusted by the regime-dependent intercept terms and variances. However, the vector equilibrium correction mechanism and the errors from the regime shifts would allow for capturing possible shifts in the equilibrium relationships between provincial employment levels and the reference economy. This would