Abstract This paper estimates **Markov** **switching** **models** with daily happiness (GNH) data from New
Zealand for a period inclusive of the Covid-19 global health pandemic. This helps us understand the dynamics of happiness due to an external shock and provides valuable information about its future evolution. Furthermore, we determine the probabilities to transition between states of happiness and estimate the duration in these states. In addition, as maximising happiness is a policy priority, we determine the factors that increase happiness, especially during the pandemic to ensure rapid restoration of happiness levels post the Covid-19 shock. The results show New Zealand is currently in an unhappy state which is lasting longer than predicted. To increase the happiness levels to pre-pandemic levels, policymakers could allow free mobility, create economic stimuli, and allow international travel between New Zealand and low- risk Covid-19 countries.

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5.3.2 A closer look at the Great Recession
Revisions to macroeconomic data are substantial (see e.g. Croushore and Stark (2001)). Using data available at the time the forecasts were made is therefore critical to evaluate realistically the **models**’ forecasting ability. Real-time employment data are available for all 50 states starting from the June 2007 vintage with last observation for May 2007. Hence, our first estimation sample extends from February 1960 to May 2007, and it is recursively expanded until August 2013. As a result, the evaluation sample extends from May 2007 to August 2013, that is 76 months. Note also that we use a real-time data series for the NBER recession dummy variable when calculating **models**’ weights so as to carefully reflect the information available at the time the forecasts were calculated. In this purely real-time experiment, since our evaluation sample covers only a limited period of time and only one recession, we do not calculate QPS statistics, but instead report the probability of being in a recession - defined as the last estimate available for the probability of being in a recession averaged across the different **Markov**-**switching** **models** (i.e., P (S t = 0|ψ t ) where t is the

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In model [4], the influence of past interest rates on present ones varies according to whether the market was in state 0 or 1 at that past date; on the other hand, in model [4’], the influence of past information depends on the state the market is in at the present date. Does it make more sense to consider that what happened in the past affects today’s behaviour according to the state the economy today (Model 4’), or according to the state of the economy at the date at which that past information was generated (Model 4)? It is not easy to decide on a priori theoretical grounds which option is better 6 . It is wise to let the data tell which version is better. Beyaert and Perez-Castejón (2000) apply Schwartz information criterion to discriminate between [4] and [4’] on the same data (except that in this paper contains three additional years of information). They obtain a systematic and overwhelming dominance of version [4] over version [4’] for p=1,2,3 and 4. This strong dominance, together with the fact that estimating these **models** is not a straightforward task, justifies that we centre on model [4] in the rest of the paper.

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available only after the global financial crisis.
Figure 9 .d displays similar results when looking at a MS model that tracks switches in both the mean and variance of the financial stress metric. The ranking of both mod- els, however, is reversed for early signals, i.e., six quarters ahead or more, with an earlier signalling ability of the logit model. The difference between the two **models** is not too sur- prising. The logit model predicts only episodes of elevated financial stress. This is closest to an MS model that tracks only changes in the mean of financial stress. Conversely, for the MS model with a **switching** variance, periods of stress are not only defined by the level of financial stress, but also by its variance, and hence the results are less com- parable with those from the logit model. In addition, for predictions several quarters ahead, the MS model tends to issue more false alarms than the logit model. Again, this is not surprising, since the MS model uses the entire distribution of the continuous stress metric to distinguish between periods of low and high financial stress, while the signalling ability is evaluated, in the end, based on the identified binary regimes representing only those episodes corresponding to the 90th percentile of the CLIFS distribution. Thus, by nature, the AUROC, if anything, is rather biased toward the predictions issued by the binary logit model.

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precisely at the time when the first countries started to recover from the global financial crisis (Figure 7 ).
However, does it mean that the probability of facing financial stress in the next quarter recovered from the MS model would have failed to correctly identify episodes of high financial stress before 2008? Figure 8 shows the one-step ahead probability of high financial stress obtained from the MS for each of the 15 EU countries. The solid blue line corresponds to the probability computed in-sample for our benchmark model including the main leading indicators discussed above. In addition, the red dashed line represents the out-of-sample one-step ahead probability computed recursively from 2006Q4 onwards by adding one quarter of new information at a time. Both lines show a surprisingly similar pattern despite the high uncertainty involved in the estimation of the **Markov** chain. We interpret this result as evidence that the main output of the MS model, i.e. the one-step ahead probability of high financial stress, is relatively consistent over time. Even using only data prior to the occurrence of the global financial crisis, the model would have successfully identified periods of low and high financial stress before the onset of the crisis. However, our results also suggest that some apparent early warning signals issued already in 2006 were obtained from using information which only became available later. For example, the increase in the in-sample probability of financial stress (solid blue line) for Spain is not visible when considering the out-of-sample probability (dashed red line) which uses only information available at that time. Overall, while the MS model consistently identifies periods of low and high financial stress irrespective of including or excluding the global financial crisis in the sample, the early warning properties of some indicators appear much more limited when considering an out-of-sample exercise.

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This article aims at highlighting diﬀerences and similarities between developed and developing countries regarding their vulnerability, observed in this article as the diﬀerent characteristics related to recessions such as the probability of entering or exiting a re- cession and the average loss encountered during a contraction. This study focuses on 80 countries at a quarterly frequency, which is a much wider approach than those found in the economic literature. The data used consist only of quarterly real GDP and popula- tion. In a first attempt to characterize countries’ vulnerability, recessions are determined using a simple dating algorithm. This approach identifies that, contrarily to what is usually said in the literature, developing countries face the same pattern as developed ones, with a probability to enter a recession around 5% and a probability to exit a re- cession around 18%. Developing countries form however an heterogeneous group. Latin American countries tend to be more vulnerable than others (i.e. with higher entrance probabilities), while Asian countries perform on average even better than developed ones. This echoes the main discussions on their regional performances across past decades, with Latin American countries having encountered several crises while Asian countries have been acknowledged for their strong growth performances. The main diﬀerence between the groups is the fact that developing countries tend to lose around twice as much output as developed ones during a contraction. This observation is valid for all developing coun- tries’ groups, thus underlying a higher vulnerability. This article then uses a nonlinear parametric approach, a **Markov**-**Switching** Model, to study the countries. If some coun- tries have less observations than other countries (which is particularly true for African countries), and thus lead to surprising results, the rest of the results represent faithfully the patterns identified before. The **Markov**-**Switching** Model proves to be a useful tool in identifying the recessions and the estimations echoe the previously presented pattern. The estimations derived using the cyclical component of the series, computed using the HP filter, help extending the definition of a recession by considering not only negative growth episodes but all episodes that imply a significantly lower growth than the trend. These results retranscribe close values to those mentioned earlier (an entrance probability around 5,5% and an exit probability around 23,3%) and identify the same observations. Moreover, using a Noise-to-Signal ratio to compare both methodology, we find values under 30%, which comforts the use of MSM estimates in identifying recessions.

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A final remark is necessary with regard to the type of clustering chosen. In this paper a hard clustering approach is used, i. e. each time-series belongs to one and only one cluster at a given time. An alternative would be soft clustering, where cluster membership is represented proba- bilistically. An example of the latter approach are finite mixture **models**. In the absence of panel data it is well established that soft clustering is preferable to hard clustering. By construction, the probabilistic assignment to clusters allows an assessment of the confidence of the cluster assign- ments. More importantly, in the absence of panel data hard clustering has been shown to lead to reasonable clusters, but inconsistent parameter estimates (Celeux and Govaert, 1993; Bryant, 1991; McLachlan, 1982). Soft clustering in the context of **Markov** **switching** **models** is possible (Butler, 2003; Alon et al., 2003; Wichern, 2001; Cadez and Heckermann, 2003), but computationally very demanding and rarely used. In the context of the proposed model a further difficulty would arise: The **Markov** **switching** clusters in this paper differ only with respect to the transition probabilities, but not with respect to the state coefficients. This implies that by construction the differences in the incomplete/complete-data log-likelihood functions tend to be small so that the traditional smoothed model probabilities, i. e. the probability that given the parameters the data has been generated by cluster m, are too close to each other to allow for a soft clustering mechanism to be well defined. Despite opting for the hard clustering approach, our model does not suffer from the inconsistency problem pointed out in the hard clustering of mixture **models**. Since panel data is available the cluster assignment is consistent for large enough time-series. If the cluster assignment is consistent, so are the parameter estimates. 24 However, since each time-series is deterministically

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Abstract
We examine price discovery in the Credit Default Swap and cor- porate bond market. By using a **Markov** **switching** framework we are able to analyze the dynamic behavior of the information shares dur- ing tranquil and crisis periods. The results show that price discovery takes place mostly on the CDS market. The importance of the CDS market even increases during the more volatile crisis periods. Accord- ing to a cross sectional analysis liquidity is the main determinant of a market’s contribution to price discovery. During the crisis period, however, we also find a positive link between leverage and CDS market information shares. Overall the results indicate that price discovery measures and their determinants change during tranquil and crisis pe- riods, which emphasizes the importance of more flexible frameworks, such as **Markov** **switching** **models**.

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1 Introduction
**Markov** regime **switching** **models** have been widely applied in economics and finance. Since the seminal application of Hamilton (1989) to U.S. real Gross National Product growth and the well-known NBER business cycle classification, the model has been adopted in numerous other applications. Examples are switches in the level of a time series, switches in the (autoregressive) dynamics of vector time series, switches in volatilities, and switches in the correlation or dependence structure between time series; see Hamilton and Raj (2002) for a partial survey. The key attractive feature of **Markov** **switching** **models** is that the conditional distribution of a time series depends on an underlying latent state or regime, which can take only a finite number of values. The discrete state evolves through time as a discrete **Markov** chain and we can summarize its statistical properties by a transition probability matrix.

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In this paper a Taylor rule, which includes the exchange rate gap, is estimated for Switzerland. In order to account for the state-contingent nature of the SNB’s monetary policy, the Taylor rule is enhanced with parameters depending on two states governed by a **Markov** **switching** process. An attractive feature of **Markov** **switching** **models** is that no restrictions regarding the size, sign, or the state at a particular point in time have to be imposed on the parameters in estimation, but are all determined by the data.

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In this paper we extend the DFMS model with the aim of accelerating peak dating. With this purpose in mind, we allow the probability to switch from an expansion to a contraction phase to be time-varying. To bring about such time-variation we propose an autoregressive structure driven by endogenous information in the form of the log-likelihood score and additionally by exogenous variables. This framework is therefore a multivariate extension of the methods described by Bazzi et al. (2017) for **Markov**-**switching** **models** for univariate time series. Furthermore, we make use of the accelerated score-driven method recently introduced by Blasques et al. (2019) and tailor it to our needs. In this approach the magnitude of the parameter update for a given value of the score-function is also made time-varying. The resulting accelerated Generalized Autoregressive Score with eXogenous variables (aGASX) model thus combines the ideas of exogenous and endogenous drivers of the transition probabilities from Diebold et al. (1994) and Durland and McCurdy (1994), respectively.

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University of Kiel August 20, 2018
Abstract
Nonlinear, non-Gaussian state space **models** have found wide applications in many areas. Since such **models** usually do not allow for an analytical representa- tion of their likelihood function, sequential Monte Carlo or particle filter methods are mostly applied to estimate their parameters. Since such stochastic approxima- tions lead to non-smooth likelihood functions, finding the best-fitting parameters of a model is a non-trivial task. In this paper, we compare recently proposed itera- tive filtering algorithms developed for this purpose with simpler online filters and more traditional methods of inference. We use a highly nonlinear class of **Markov**- **switching** **models**, the so called **Markov**-**switching** multifractal model (MSM), as our workhorse in the comparison of different optimisation routines. Besides the well-established univariate discrete-time MSM, we introduce univariate and mul- tivariate continuous-time versions of MSM. Monte Carlo simulation experiments indicate that across a variety of MSM specifications, the classical Nelder-Mead or simplex algorithm appears still as more efficient and robust compared to a number of online and iterated filters. A very close competitor is the iterated filter recently proposed by Ionides et al. (2006) while other alternatives are mostly dominated by these two algorithms. An empirical application of both discrete and continuous-time MSM to seven financial time series shows that both **models** dominate GARCH and FIGARCH **models** in terms of in-sample goodness-of-fit. Out-of-sample forecast comparisons show in the majority of cases a clear dominance of the continuous-time MSM under a mean absolute error criterion, and less conclusive results under a mean squared error criterion.

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3 The **Markov** **switching** SVAR model
The identification through heteroscedasticity is a powerful option to support iden- tification of shocks in SVAR **models**, see Rigobon (2003) or Lanne and Lütkepohl (2008), among others. In comparison to classical identifying techniques like short run, long run or sign restrictions, the identification through heteroscedasticity is a more data oriented approach. This is also in sharp contrast to the identification strategies for latent dynamic factor **models**, previously applied to inflation series in e.g. Mumtaz and Surico (2012), where factor loadings are restricted to zero such that country specific factors are easily characterized by having no impact on foreign in- flation. With the SVAR model and the identification through heteroscedasticity, we attempt to explore transmission channels and the economic nature of driving forces more deeply. We let the data speak about the statistical identification and check in a second step whether some economic meaning can be attached to the individual structural shocks. In general, our statistical procedure allows some country specific shock to transmit to foreign inflation expectations.

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Unfortunately, this way of composing **Markov** chains is very uninteresting. Our assumption that the component CTMCs are independent, means that the corresponding components of the system do not influence each other, so we are bound to complex system **models** where components are completely isolated. But for those systems we can model and study the components in isolation. There is no point in composing them in the first place, if they do not interact in some way or another. What we are missing is a way to represent the fact that components in a system may indeed interact. To put it differently, we need a way to model that the behaviour of one component in a system may be influenced (changed) by the behaviour of other components in the same system. In this thesis, we will attack this problem by combining CTMCs with a purely interaction-oriented discrete-state formalism.

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Results reveal that the Central Bank of Brazil placed a heavier weight on inflation stabilization to the detriment of output and exchange rate throughout the analyzed period. As no regime shifts were found in Taylor rule parameters, this paper confirms that studies on the Taylor rule for the Brazilian economy that do not contemplate regime **switching** are on the right path. Nevertheless, the **models** identified three periods of high volatility in Brazil: (i) from 1998 to 2000; (ii) throughout 2003; and (iii) from 2008 to 2010, period of the U.S. crisis, with some differences as to its beginning and end. Thus, the change in exogenous shock parameters should be taken into account in **models** for the Brazilian economy. In addition, the Phillips curve parameters were not constant over time (model 2).

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Suggested Citation: Fella, Giulio; Gallipoli, Giovanni; Pan, Jutong (2017) : **Markov**-chain
approximations for life-cycle **models**, Working Paper, No. 827, Queen Mary University of London, School of Economics and Finance, London
This Version is available at: http://hdl.handle.net/10419/184778

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Simulation results. The complete set of tables and figures can be obtained on inquiry to the author. Here only a few of them are presented. As we can see from the results, our algorithm was in many cases faster than the EM algorithm. The differences in BIC were in the most cases not significant, as the p-values from the corresponding Wilcoxon tests suggest. An exception was the SEM algorithm for **Models** 2 and 3 - in the cases where SEM converged, it achieved a significantly better BIC values than the rest. The only problem was, that such cases were quite rare (see Table 1.1), so we could achieve the same effect for any other algorithm by considering e.g. only the best 30% of results.

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The early works in population genetics (Fisher, 1930, in particular) have inspired still another modeling approach that is related to ABMs, namely, evolutionary game theory (see Smith, 1982 for a seminal volume and Roca et al., 2009 for a recent review). Here, games are designed in which agents repeatedly play against one another adopting one out of a set of predefined strategies. A fitness is assigned to the combinations of strategies and the population evolves as a response to this fitness. As in the framework of statistical mechanics, the model evolution is typically captured in form of differential equation describing the evolution of the (relative) frequencies of the different strategies, referred to as replicator dynamics in this context (Taylor and Jonker, 1978; Schuster and Sigmund, 1983; Hofbauer and Sig- mund, 2003). One of the main purposes of this work is to spell out explicitly how to link the dynamics at the micro level to these macroscopic descriptions. Finally, it is worth mentioning that research in economics has experi- enced a growing interest in modeling economic phenomena as the result of the interactions of heterogeneous individuals (Tesfatsion and Judd, 2006). In particular in the field of finance, this has led to the development of ABMs for the identification of (macro) patterns of collective dynamics from (mi- cro) investor heterogeneity in many financial settings (Cont and Bouchaud, 2000; LeBaron, 2000; Hommes, 2006; Preis et al., 2013). Noteworthy, there is also a number of empirical applications of **Markov** chains in the field of finance (e.g., Corcuera et al., 2005; Nielsen, 2005; Norberg, 2006). Interac- tion and heterogeneity on the one hand, and non-Gaussianity, heavy tails and long-range correlations on the other appear to be natural features of modern economies, to which the formerly dominating tradition of modeling representative agents has, to a large extent, paid little attention. This thesis shows that memory effects at the macroscopic level are an immediate conse- quence of microscopic heterogeneity and it may therefore contribute to the identification of the relevant microscopic mechanisms that presumably play a role in the market.

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Given the above, the question arises as to what has triggered the occasional scepticism over the suitability and/or sustainability of Hong Kong’s currency board system? Since the inception of the global financial crisis of 2008 – 2009, which brought financial markets into turmoil, we now have extensive theoretical research suggesting that the pricing of assets, including exchange rates, may be non-linear. Recent papers have stressed the importance of non-linear effects and amplification dynamics during financial crises. The theory suggests that relatively small shocks can have large spillover effects [Brunnermeier and Pedersen (2008)]. Moreover, Brock et al. (2009) have shown that hedging instruments may produce non-linear dynamics and destabilize markets. Bianchi (2011) and Jermann and Quadrini (2012) have formalised the idea of a regime-dependent role of financial markets. Looking at exchange rates, Jeanne and Masson (2000) have addressed sunspot-driven multiple equilibria in the exchange rate context. They prove that the effects of sunspot shocks are absorbed by discrete jumps in the intercept of a regression of the currency devaluation probability on fundamental variables. Therefore, a **Markov** regime-**switching** test can be used to identify sunspot equilibria. An alternative theory for regime-**switching** uses the “animal spirits” concept of De Grauwe (2010) and De Grauwe and Kaltwasser (2012). Here, boundedly rational and imperfectly informed agents use heuristics to make decisions in the foreign exchange market. Again, agents’ psychological movements are self-fulfilling, as waves of optimism and pessimism lead to fluctuations of the exchange rate even when the underlying fundamentals are unaltered by an exogenous shock. However, it should be noted that different authors point to a variety of causal mechanisms. A number of studies have examined the idea of regime-**switching** credibility in exchange rate dynamics. See, for example, Sarantis and Piard (2004), Arestis and Mouratidis (2005), Chen (2006) and Altavilla and De Grauwe (2010). One way to capture (albeit in a reduced-form way) the impact of financial factors shaping credibility is to employ **Markov**-**switching** VAR (MS-VAR) **models** with time-varying transition probabilities. In our view, such **models** have much to contribute and offer us a promising avenue of empirical research. 5

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