Engle granger cointegration em stata forex

Base R navios com um monte de funcionalidade útil para séries de tempo, em especial no pacote stats. Isto é complementado por muitos pacotes no CRAN, que são brevemente resumidos abaixo. Há também uma considerável sobreposição entre as ferramentas para séries temporais e aquelas nas vistas de tarefas de Econometria e Finanças. Os pacotes nesta visualização podem ser estruturados grosso modo nos seguintes tópicos. Se você acha que algum pacote está faltando na lista, entre em contato conosco. A infraestrutura . Base R contém infra-estrutura substancial para representar e analisar dados de séries temporais. A classe fundamental é quottsquot que pode representar séries temporais regularmente espaçadas (usando carimbos de tempo numéricos). Assim, é particularmente adequado para dados anuais, mensais, trimestrais, etc. As médias móveis são computadas por ma da previsão. E rollmean do jardim zoológico. Este último também proporciona uma função geral rollapply. Juntamente com outras funções de estatísticas de rolamento específicas. Roll fornece funções paralelas para computar estatísticas de rolagem. Gráficos . As parcelas de séries temporais são obtidas com plot () aplicado aos objetos ts. (Parciais) funções de autocorrelação são implementadas em acf () e pacf (). Versões alternativas são fornecidas por Acf () e Pacf () na previsão. Juntamente com uma exibição de combinação usando tsdisplay (). SDD fornece diagramas mais gerais de dependência serial, enquanto dCovTS calcula e traça as funções de covariância e correlação de distância de séries temporais. Exibições sazonais são obtidas usando monthplot () nas estatísticas e seasonplot na previsão. Wats implementa wrap-around gráficos de série de tempo. Ggseas fornece gráficos ggplot2 para séries ajustadas sazonalmente e estatísticas de rolamento. Dygraphs fornece uma interface para o Dygraphs interativo série de tempo gráfico de biblioteca. ZRA traça objetos de previsão do pacote de previsão usando dígrafos. As parcelas dos ventiladores básicos das distribuições de previsão são fornecidas por previsão e vars. As parcelas de ventiladores mais flexíveis de quaisquer distribuições seqüenciais são implementadas em fanplot. Class quottsquot só pode lidar com carimbos de tempo numéricos, mas muitas mais classes estão disponíveis para armazenar informações de data / hora e computação com ele. Para uma visão geral, consulte R Help Desk: Classes de data e hora em R de Gabor Grothendieck e Thomas Petzoldt em R News 4 (1). 29-32. Classes quotyearmonquot e quotyearqtrquot do zoológico permitem computação mais conveniente com observações mensais e trimestrais, respectivamente. Classe quotDatequot do pacote base é a classe básica para lidar com datas em dados diários. As datas são internamente armazenadas como o número de dias desde 1970-01-01. O pacote cron fornece classes para datas (). Horas () e data / hora (intra-dia) em cron (). Não há suporte para fusos horários e horário de verão. Internamente, os objetos quotchronquot são dias (fracionários) desde 1970-01-01. Classes quotPOSIXctquot e quotPOSIXltquot implementam o padrão POSIX para informações de data / hora (intra-dia) e também suportam fusos horários e horário de verão. No entanto, os cálculos de fuso horário exigem algum cuidado e podem depender do sistema. Internamente, quotPOSIXctquot objetos são o número de segundos desde 1970-01-01 00:00:00 GMT. O pacote lubridate fornece funções que facilitam determinados cálculos baseados em POSIX. A classe quottimeDatequot é fornecida no pacote timeDate (anteriormente: fCalendar). Destina-se a informações de data / hora financeira e trata de fusos horários e horários de verão através de um novo conceito de centros financeiros. Internamente, ele armazena todas as informações em quotPOSIXctquot e faz todos os cálculos em GMT apenas. Funcionalidade do calendário, p. Incluindo informações sobre fins de semana e feriados para várias bolsas de valores, também está incluído. O pacote tis fornece a classe quottiquot para informações de hora / data. A classe quotmondatequot do pacote mondate facilita a computação com datas em termos de meses. O pacote tempdisagg inclui métodos para desagregação temporal e interpolação de uma série temporal de baixa freqüência para uma série de freqüências mais altas. A desagregação das séries cronológicas também é fornecida por tsdisagg2. TimeProjection extrai componentes de tempo útil de um objeto de data, como dia da semana, fim de semana, feriado, dia do mês, etc, e colocá-lo em um quadro de dados. Como mencionado acima, quotts ​​é a classe básica para séries temporais regularmente espaçadas usando carimbos de tempo numéricos. O pacote do zoológico fornece infra-estrutura para séries temporais regular e irregularmente espaçadas usando classes arbitrárias para os carimbos de tempo (ou seja, permitindo que todas as classes da seção anterior). Ele é projetado para ser o mais consistente possível com quottsquot. Coerção de e para quotzooquot está disponível para todas as outras classes mencionadas nesta seção. O pacote xts é baseado no zoológico e fornece um tratamento uniforme de classes de dados Rs diferentes baseadas no tempo. Vários pacotes implementam séries temporais irregulares baseadas em quotPOSIXctquot selos de tempo, destinados especialmente para aplicações financeiras. Estes incluem quotits de sua. Citações de tseries. E quotftsquot de fts. A classe quottimeSeriesquot em timeSeries (anteriormente: fSeries) implementa séries de tempo com selos de tempo quottimeDatequot. A classe quottisquot in tis implementa séries de tempo com selos de tempo quottiquot. O pacote tframe contém infra-estrutura para definir intervalos de tempo em diferentes formatos. Previsão e Modelagem Univariada O pacote de previsão fornece uma classe e métodos para previsões de séries de tempo univariadas e fornece muitas funções implementando diferentes modelos de previsão incluindo todos aqueles no pacote de estatísticas. Suavização exponencial. HoltWinters () em estatísticas fornece alguns modelos básicos com otimização parcial, ets () do pacote de previsão fornece um conjunto maior de modelos e instalações com otimização completa. Robets fornece uma alternativa robusta à função ets (). Smooth implementa algumas generalizações de suavização exponencial. O pacote MAPA combina modelos exponenciais de suavização em diferentes níveis de agregação temporal para melhorar a precisão das previsões. O método theta é implementado na função thetaf do pacote de previsão. Uma implementação alternativa e estendida é fornecida no forectheta. Modelos auto-regressivos. Ar () em estatísticas (com seleção de modelo) e FitAR para modelos de subconjunto AR. Modelos ARIMA. Arima () em stats é a função básica dos modelos ARIMA, SARIMA, ARIMAX e subconjunto ARIMA. Ele é aprimorado no pacote de previsão através da função Arima () juntamente com auto. arima () para a seleção de ordem automática. Arma () no pacote tseries fornece algoritmos diferentes para modelos ARMA e subconjunto ARMA. O FitARMA implementa um algoritmo MLE rápido para modelos ARMA. O pacote gsarima contém funcionalidade para a simulação da série de tempo SARIMA generalizada. O pacote mar1s processa AR multiplicativo (1) com processos sazonais. TSTutorial fornece um tutorial interativo para Box-Jenkins modelagem. Os intervalos de previsão melhorados para ARIMA e modelos de séries temporais estruturais são fornecidos pelo tsPI. Modelos ARMA periódicos. Pear e partsm para modelos periódicos de séries temporais autorregressivas e perARMA para modelagem periódica ARMA e outros procedimentos para análise periódica de séries temporais. Modelos ARFIMA. Algumas instalações para modelos ARFIMA fraccionados diferenciados são fornecidas no pacote fracdiff. O pacote arfima possui recursos mais avançados e gerais para modelos ARFIMA e ARIMA, incluindo modelos de regressão dinâmica (função de transferência). ArmaFit () do pacote fArma é uma interface para modelos ARIMA e ARFIMA. Ruído gaussiano fracionário e modelos simples para séries de tempo de decaimento hiperbólico são tratados no pacote FGN. Os modelos de função de transferência são fornecidos pela função arimax no pacote TSA ea função arfima no pacote arfima. Outlier detecção após a Chen-Liu abordagem é fornecida por tsoutliers. Modelos estruturais são implementados em StructTS () em stats, e em stsm e stsm. class. KFKSDS fornece uma implementação ingênua do filtro de Kalman e alisadores para modelos de espaço de estados univariados. Os modelos de séries temporais estruturais bayesianas são implementados em séries temporais não Gaussianas podem ser manipuladas com modelos de espaço de estado GLARMA via glarma. E usando modelos Generalized Autoregressive Score no pacote GAS. Modelos condicionais de Auto-Regressão utilizando métodos Monte Carlo Likelihood são implementados em mclcar. Modelos GARCH. Garch () do tseries se encaixa modelos GARCH básicos. Muitas variações em modelos de GARCH são fornecidas pelo rugarch. Outros pacotes GARCH univariados incluem fGarch que implementa modelos ARIMA com uma ampla classe de inovações GARCH. Existem muitos outros pacotes GARCH descritos na vista de tarefas Financeiro. Modelos de volatilidade estocástica são manipulados por stochvol em uma estrutura bayesiana. Modelos de série de tempo de contagem são manipulados nos pacotes tscount e acp. O ZIM fornece modelos Zero-Inflated para séries de tempo de contagem. Tsintermittent implementa vários modelos para analisar e prever séries de tempo de demanda intermitente. As séries temporais censuradas podem ser modeladas usando centavos e carx. Testes Portmanteau são fornecidos através de Box. test () no pacote stats. Testes adicionais são dados por portes e WeightedPortTest. A detecção de ponto de mudança é fornecida no strucchange (usando modelos de regressão linear), na tendência (usando testes não paramétricos) e em wbsts (usando segmentação binária selvagem). O pacote de ponto de mudança fornece muitos métodos de ponto de troca populares, e ecp faz detecção de ponto de mudança não paramétrico para séries univariadas e multivariadas. A detecção de ponto de mudança online para séries temporais univariadas e multivariadas é fornecida pelo onlineCPD. O InspectChangepoint usa projeção esparsa para estimar pontos de mudança em séries temporais de alta dimensão. A imputação de séries temporais é fornecida pelo pacote imputeTS. Alguns recursos mais limitados estão disponíveis usando na. interp () do pacote de previsão. As previsões podem ser combinadas usando ForecastCombinations que suporta os métodos mais usados ​​para combinar previsões. ForecastHybrid fornece funções para previsões de conjuntos, combinando abordagens do pacote de previsão. O GeomComb fornece métodos de combinação de previsão baseados em eigenvector (geométricos), bem como outras abordagens. Opera tem facilidades para previsões on-line com base em combinações de previsões fornecidas pelo usuário. A avaliação da previsão é fornecida na função accuracy () da previsão. A avaliação de previsão distribucional usando regras de pontuação está disponível no scoringRules Miscellaneous. Ltsa contém métodos para análise de séries temporais lineares, timsac para análise e controle de séries temporais e tsbugs para modelos BUGS de séries temporais. A estimativa da densidade espectral é fornecida pelo espectro () no pacote stats, incluindo o periodograma, o periodograma suavizado e as estimativas de AR. A inferência espectral bayesiana é fornecida por bspec. O quantspec inclui métodos para calcular e traçar periodogramas de Laplace para séries temporais univariadas. O periodograma Lomb-Scargle para séries temporais amostradas de forma desigual é calculado por lomb. O espectro utiliza transformadas de Fourier e Hilbert para filtragem espectral. Psd produz estimativas de densidade espectral adaptativas, seno-multitaper. Kza fornece Kolmogorov-Zurbenko Adaptive Filters incluindo detecção de quebra, análise espectral, wavelets e KZ Fourier transformações. Multitaper também fornece algumas ferramentas de análise espectral multitaper. Métodos Wavelet. O pacote de wavelets inclui computar filtros wavelet, transformadas wavelet e análises multiresolução. Os métodos wavelet para a análise de séries temporais baseados em Percival e Walden (2000) são dados em wmtsa. WaveletComp fornece algumas ferramentas para a análise baseada em wavelet de séries temporais univariadas e bivariadas incluindo cross-wavelets, diferença de fase e testes significativos. Biwavelet pode ser usado para plotar e calcular os espectros wavelet, espectros de wavelet cruzado e coerência wavelet de séries temporais não-estacionárias. Ele também inclui funções para agrupar séries temporais baseadas nas (des) similaridades em seu espectro. Testes de ruído branco usando wavelets são fornecidos por hwwntest. Métodos Wavelet adicionais podem ser encontrados no brainwaver pacotes. Rwt. Waveslim Wavethresh e mvcwt. A regressão harmônica usando termos de Fourier é implementada em HarmonicRegression. O pacote de previsão também fornece algumas facilidades simples de regressão harmônica através da função fourier. Decomposição e filtragem Filtros e suavização. O filtro () em stats fornece filtragem linear média auto-regressiva e móvel de séries temporais univariadas múltiplas. O pacote robfilter fornece vários filtros robustos de séries temporais, enquanto o mFilter inclui diversos filtros de séries temporais úteis para suavizar e extrair componentes tendenciais e cíclicos. Smooth () do pacote de estatísticas computa Tukeys executando mediana smoothers, 3RS3R, 3RSS, 3R, etc sleekts calcula o 4253H duas vezes método de suavização. Decomposição. A decomposição sazonal é discutida abaixo. A decomposição baseada em auto-regressão é fornecida pelo ArDec. O rmaf usa um filtro refinado de média móvel para decomposição. Análise de Espectro Singular é implementada em Rssa e métodos espectrais. A decomposição do modo empírico (EMD) ea análise espectral de Hilbert são fornecidas por EMD. Ferramentas adicionais, incluindo EMD conjunto, estão disponíveis em hht. Uma implementação alternativa do conjunto EMD e sua variante completa estão disponíveis em Rlibeemd. Decomposição sazonal. O pacote stats fornece decomposição clássica em decompose (). E decomposição STL em stl (). A decomposição STL melhorada está disponível em stlplus. StR fornece a decomposição Seasonal-Trend baseada na regressão. X12 fornece um wrapper para os binários X12 que devem ser instalados primeiro. X12GUI fornece uma interface gráfica do usuário para x12. Os binários X-13-ARIMA-SEATS são fornecidos no pacote x13binary, com o fornecimento sazonal de uma interface R. Análise da sazonalidade. O pacote bfast fornece métodos para detectar e caracterizar mudanças abruptas dentro da tendência e componentes sazonais obtidos a partir de uma decomposição. Npst fornece uma generalização do teste de sazonalidade Hewitts. estação. Análise sazonal de dados de saúde, incluindo modelos de regressão, crossover caso-estratificado tempo, funções de traçado e verificações residuais. Mares Análise sazonal e gráficos, especialmente para climatologia. Dessazonalizar Otimização da dessazonalização para séries geofísicas usando o encaixe AR. Estacionaridade, Unidade de Raízes e Cointegração Estacionaridade e raízes unitárias. Tseries fornece vários testes de estacionaridade e raiz unitária, incluindo Dickey-Fuller aumentado, Phillips-Perron e KPSS. Implementações alternativas dos testes ADF e KPSS estão no pacote urca, que também inclui métodos adicionais como os testes Elliott-Rothenberg-Stock, Schmidt-Phillips e Zivot-Andrews. O pacote fUnitRoots também fornece o teste MacKinnon, enquanto o uroot fornece testes de raiz unitária sazonais. O CADFtest fornece implementações tanto do ADF padrão como de um teste de ADF (CADF) com covariável. Estacionaridade local. Localiza um teste de estacionaridade local e calcula a autocovariância localizada. A determinação da determinação da costeração de séries temporais é fornecida por costat. LSTS tem funções para análise de séries temporais localmente estacionárias. Modelos de wavelet estacionariamente estacionários para séries temporais não-estacionárias são implementados em wavethresh (incluindo estimativa, plotagem e simulação para espectros que variam no tempo). Cointegração. O método Engle-Granger de dois passos com o teste de cointegração Phillips-Ouliaris é implementado em tseries e urca. Este último contém adicionalmente funcionalidade para os testes Johansen trace e lambda-max. TsDyn fornece Johansens teste e AIC / BIC seleção simultânea rank-lag. O CommonTrend fornece ferramentas para extrair e traçar tendências comuns de um sistema de cointegração. A estimação de parâmetros ea inferência em uma regressão de cointegração são implementadas em cointReg. Análise não linear de séries temporais Auto-regressão não-linear. Várias formas de auto-regressão não linear estão disponíveis em tsDyn incluindo AR aditivo, redes neurais, modelos SETAR e LSTAR, limiar VAR e VECM. A auto-regressão da rede neural também é fornecida no GMDH. O bentcableAR implementa a autorregressão Bent-Cable. BAYSTAR fornece a análise bayesiana de modelos autorregressivos de limiar. TseriesChaos fornece uma implementação R dos algoritmos do projeto TISEAN. Autoregression Os modelos de comutação de Markov são fornecidos em MSwM. Enquanto que as misturas dependentes de modelos de Markov latentes são dadas em depmix e depmixS4 para séries temporais categóricas e contínuas. Testes. Vários testes de não-linearidade são fornecidos em fNonlinear. TseriesEntropy testes de dependência serial não linear com base em métricas de entropia. Funções adicionais para séries temporais não-lineares estão disponíveis em nlts e nonlinearTseries. A modelagem da série de tempo do Fractal e a análise são fornecidas pelo fractal. Fractalrock gera séries de tempo fractal com distribuições de retornos não-normais. Modelos dinâmicos de regressão Modelos dinâmicos lineares. Uma interface conveniente para ajustar modelos de regressão dinâmica via OLS está disponível no dynlm uma abordagem avançada que também funciona com outras funções de regressão e mais séries de séries temporais é implementada em dyn. Equações mais avançadas do sistema dinâmico podem ser montadas usando dse. Os modelos de espaço de estados lineares gaussianos podem ser montados usando dlm (via máxima verossimilhança, filtragem / alisamento de Kalman e métodos bayesianos), ou usando bsts que usa MCMC. As funções para a modelação não-linear de atraso distribuído são fornecidas em dlnm. Modelos de parâmetros variáveis ​​no tempo podem ser ajustados usando o pacote tpr. OrderedLasso ajusta um modelo linear esparso com uma restrição de ordem sobre os coeficientes, a fim de lidar com regressores defasados ​​onde os coeficientes decadência como o lag aumenta. Modelos dinâmicos de vários tipos estão disponíveis em dynr incluindo tempo discreto e contínuo, modelos lineares e não-lineares e diferentes tipos de variáveis ​​latentes. Modelos de séries temporais multivariadas Modelos VAR (Vector Autoregressive) são fornecidos via ar () no pacote stats básico incluindo a seleção de pedidos via AIC. Estes modelos são restritos para ser estacionário. O MTS é um conjunto de ferramentas para análise de séries temporais multivariadas, incluindo VAR, VARMA, VARMA sazonal, modelos VAR com variáveis ​​exógenas, regressão multivariada com erros de séries temporais e muito mais. Possivelmente modelos VAR não estacionários são montados no pacote mAr, o que também permite modelos VAR no espaço do componente principal. Sparsevar permite a estimação de modelos VAR e VECM esparsos, ecm fornece funções para a construção de modelos VECM, enquanto BigVAR estima modelos VAR e VARX com penalidades laço estruturado. Os modelos e redes VAR automatizados estão disponíveis no autovarCore. Modelos mais elaborados são fornecidos em pacotes vars. TsDyn. EstVARXls () em dse. E uma abordagem bayesiana está disponível em MSBVAR. Outra implementação com intervalos de previsão bootstrap é dada em VAR. etp. MlVAR fornece auto-regressão vectorial multi-nível. VARsignR fornece rotinas para identificar choques estruturais em modelos VAR usando restrições de sinal. Gdpc implementa componentes principais dinâmicos generalizados. O pcdpca estende os componentes principais dinâmicos a séries temporais multivariadas periodicamente correlacionadas. Os modelos VARIMA e modelos de espaço de estado são fornecidos no pacote dse. EvalEst facilita experiências de Monte Carlo para avaliar os métodos de estimação associados. Modelos de correção de erros vetoriais estão disponíveis através da urca. Vars e pacotes tsDyn, incluindo versões com restrições estruturais e thresholding. Análise de componentes de séries temporais. A análise fatorial de séries temporais é fornecida em tsfa. O ForeCA implementa uma análise de componentes que pode ser pesquisada procurando as melhores transformações lineares que tornam uma série de tempo multivariada o mais previsível possível. PCA4TS encontra uma transformação linear de uma série de tempo multivariada dando subseries de menor dimensão que não estão correlacionadas entre si. Modelos de espaço de estados multivariados são implementados no pacote FKF (Fast Kalman Filter). Isso proporciona modelos de espaço de estados relativamente flexíveis através da função fkf (): os parâmetros de espaço de estado podem variar em função do tempo e as interceptações são incluídas em ambas as equações. Uma implementação alternativa é fornecida pelo pacote KFAS que fornece um filtro de Kalman multivariado rápido, mais suave, simulação mais suave e previsão. Ainda outra implementação é dada no pacote dlm que também contém ferramentas para converter outros modelos multivariados em forma de espaço de estado. Dlmodeler fornece uma interface unificada para dlm. KFAS e FKF. O MARSS se ajusta a modelos de estados-espaço autoregressivos multivariados restritos e não restringidos usando um algoritmo EM. Todos esses pacotes assumem que os termos de erro observacional e de estado não estão correlacionados. Os processos de Markov parcialmente observados são uma generalização dos modelos de espaço de estados multivariados lineares usuais, permitindo modelos não-Gaussianos e não-lineares. Estes são implementados no pacote pompa. Modelos de volatilidade estocástica multivariada (usando fatores latentes) são fornecidos por fatores tochvol. Análise de grandes grupos de séries temporais O agrupamento em séries temporais é implementado em TSclust. Dtwclust. BNPTSclust e pdc. TSdist fornece medidas de distância para dados de séries temporais. O jmotif implementa ferramentas baseadas na discretização simbólica de séries temporais para encontrar motivos em séries temporais e facilita a classificação de séries temporais interpretáveis. Métodos para traçar e prever coleções de séries temporais hierárquicas e agrupadas são fornecidos por hts. Ladrão usa métodos hierárquicos para conciliar previsões de séries temporais agregadas. Uma abordagem alternativa para conciliar previsões de séries temporais hierárquicas é fornecida por gtop. Ladrão Modelos de tempo contínuo Modelagem autorregressiva de tempo contínuo é fornecida em cts. Sim. DiffProc simula e modela equações diferenciais estocásticas. A simulação ea inferência para equações diferenciais estocásticas são fornecidas por sde e yuima. Bootstrapping. O pacote de inicialização fornece a função tsboot () para bootstrapping de série de tempo, incluindo bootstrap de bloco com várias variantes. Tsbootstrap () do tseries fornece bootstrapping rápido estacionário e de bloco. O bootstrap entropy máximo para a série de tempo está disponível no meboot. Timesboot calcula o CI de bootstrap para o ACF e o periodograma da amostra. O BootPR calcula os intervalos de previsão corrigidos por bias e boostrap para séries temporais autorregressivas. Dados de Makridakis, Wheelwright e Hyndman (1998) Previsão: métodos e aplicações são fornecidos no pacote fma. Dados de Hyndman, Koehler, Ord e Snyder (2008) As previsões com suavização exponencial estão no pacote expsmooth. Dados de Hyndman e Athanasopoulos (2013) Previsão: princípios e práticas estão no pacote fpp. Os dados da competição M e competição M3 são fornecidos no pacote Mcomp. Os dados da competição M4 são dados em M4comp. Enquanto a Tcomp fornece dados da Competição de Previsão de Turismo IJF 2010. Pdfetch fornece facilidades para baixar séries econômicas e financeiras de fontes públicas. Dados do portal on-line do Quandl para conjuntos de dados financeiros, econômicos e sociais podem ser consultados interativamente usando o pacote Quandl. Os dados do portal on-line do Datamarket podem ser obtidos usando o pacote rdatamarket. Os dados de Cryer e Chan (2010) estão no pacote TSA. Os dados de Shumway e Stoffer (2011) estão no pacote astsa. Dados de Tsay (2005) A análise de séries de tempo financeiras está no pacote FinTS, juntamente com algumas funções e arquivos de script necessários para trabalhar alguns dos exemplos. TSdbi fornece uma interface comum para bancos de dados de séries temporais. Fama fornece uma interface para bases de dados de séries de tempo FAME AER e Ecdat ambos contêm muitos conjuntos de dados (incluindo dados de séries temporais) de muitos livros de econometria dtw. Algoritmos dinâmicos do warping do tempo para computar e traçar alinhamentos pairwise entre séries de tempo. EnsembleBMA. Modelo Bayesiano Averaging para criar previsões probabilísticas a partir de previsões de conjunto e observações meteorológicas. Earlywarnings. Avisos iniciais sinalizam caixa de ferramentas para detectar transições críticas em eventos de séries temporais. Transforma dados de eventos extraídos por máquina em séries de tempo multivariadas agregadas regulares. FeedbackTS. Análise da direcionalidade temporal fragmentada para investigar feedback em séries temporais. LPStimeSeries pretende encontrar similaridade de padrão quotlearned para séries de tempo. MAR1 fornece ferramentas para preparar dados de séries temporais de comunidades ecológicas para modelagem de AR multivariada. Redes Rotinas para a estimativa de redes de correlação parciais esparsas de longo prazo para dados de séries temporais. PaleoTS. Modelagem da evolução em séries temporais paleontológicas. Pastecs. Regulação, decomposição e análise de séries espaço-temporais. Ptw. Comprimento de tempo paramétrico. RGENERATE fornece ferramentas para gerar séries de vetores. RMAWGEN é um conjunto de funções S3 e S4 para a geração estocástica espacial multi-site de séries de tempo diárias de temperatura e precipitação, fazendo uso de modelos VAR. O pacote pode ser utilizado em climatologia e hidrologia estatística. RSEIS. Ferramentas sísmicas de análise de séries temporais. Rts. Análise de séries temporais de rasterização (por exemplo, séries temporais de imagens de satélite). Sae2. Modelos de séries temporais para estimação de área pequena. SpTimer. Modelagem bayesiana espacial-temporal. vigilância. Modelação temporal e espaço-temporal e monitoramento de fenômenos epidêmicos. TED. Turbulência de séries temporais Detecção e classificação de eventos. Marés Funções para calcular características de séries temporais quase periódicas, e. Níveis de água nos estuários. tigre. Grupos temporariamente resolvidos de diferenças típicas (erros) entre duas séries temporais são determinados e visualizados. TSMining. Motivos Univariados e Multivariados em Dados de Série de Tempo. TsModel. Modelagem de séries temporais para poluição do ar e saúde. Modelos de taxas de inflação e taxas de câmbio no Gana: aplicação de modelos GARCH multivariados Aceito: 19 de janeiro de 2015 Publicado em: 24 de fevereiro de 2015 Este artigo teve como objetivo investigar a volatilidade e a relação condicional entre taxas de inflação, taxas de câmbio e taxas de juros Bem como construir um modelo usando modelos GARCH DCC e BEKK multivariados usando dados do Gana de janeiro de 1990 a dezembro de 2013. O estudo revelou que a depreciação acumulada do cedi para o dólar dos EUA de 1990 a 2013 é de 7.010,2 e a depreciação ponderada anual de O cedi para o dólar americano para o período é de 20,4. Havia evidências de que, o fato de a taxa de inflação ser estável, não significa que as taxas de câmbio e as taxas de juros devem ser estáveis. Em vez disso, quando o cedi executa bem no forex, taxas de inflação e taxas de juros reagem positivamente e se tornam estáveis ​​no longo prazo. O modelo BEKK é robusto para modelar e prever a volatilidade das taxas de inflação, taxas de câmbio e taxas de juros. O modelo DCC é robusto para modelar a correlação condicional e incondicional entre taxas de inflação, taxas de câmbio e taxas de juros. O modelo BEKK, que prevê alta volatilidade da taxa de câmbio para o ano de 2014, é muito robusto para modelar as taxas de câmbio no Gana. A equação média do modelo DCC também é robusta para prever as taxas de inflação no Gana. DCC BEKK GARCH Gana Volatilidade Inflação Câmbio Taxas de juros Introdução Quando o nível geral de preços é relativamente estável, as incertezas de atividades relacionadas ao tempo como o investimento diminuem. Isso ajuda a promover o pleno emprego e o forte crescimento econômico. Quando a estabilidade dos preços é alcançada e mantida, os decisores de política monetária têm feito o seu trabalho bem (Sobel et al., 2006). Concebivelmente, uma das responsabilidades mais importantes de cada governo é promover uma economia saudável, que beneficie todos os seus cidadãos. O governo através de sua capacidade de tributar, gastar e controlar a oferta de dinheiro, tenta promover o pleno emprego, a estabilidade de preços eo crescimento econômico. A importância da estabilidade de preços é igualmente sublinhada no Acordo de Maastricht, que definiu o quadro para uma moeda única europeia, e identificou a estabilidade de preços como o principal objectivo do novo Banco Central Europeu (McEachern 2006). A deflação pode resultar em desgraça para uma economia que é, enfraquece a demanda dos consumidores por bens e serviços, como as famílias provavelmente não gastam, acreditando que os preços continuarão a cair. Isto significa que as empresas, bem como o governo pode ser incapaz de pagar dívidas e poderia resultar em retração. Enfatizando este ponto, Lagarde, o Director-Geral do FMI, advertiu em Abril de 2014 para a área do euro que um período prolongado de baixa inflação ou de deflação pode suprimir a procura e a produção e anular o crescimento eo emprego. De acordo com Goldberg e Knetter (1997), a taxa de câmbio é a variação percentual dos preços de importação em moeda local resultante de uma mudança de um por cento na taxa de câmbio entre os países exportadores e importadores. A repercussão da taxa de câmbio é, portanto, o efeito (positivo ou negativo) das taxas de câmbio nos preços de importação e exportação, nos preços ao consumidor ou na inflação, nos investimentos e nos volumes comerciais. Engel e Rogers (1996) estabeleceram que o cruzamento da fronteira EUA-Canadá pode aumentar consideravelmente a volatilidade dos preços relativos e que as flutuações nas taxas de câmbio explicam cerca de um terço do aumento da volatilidade. Essa é a fronteira EUA-Canadá é um determinante importante da volatilidade dos preços relativos, mesmo depois de ter devidamente em conta o papel da distância. Parsley e Wei (2001) confirmaram resultados anteriores de que a passagem das fronteiras nacionais aumenta significativamente a dispersão dos preços. A demanda e a oferta de moeda são os principais determinantes das taxas de câmbio. Taxa de Juros Paridade é um conceito importante que explica o estado de equilíbrio da relação entre taxa de juros e taxa de câmbio de dois países. O mercado de câmbio está em equilíbrio quando os depósitos de todas as moedas oferecem a mesma taxa esperada de retorno. A condição de que os retornos esperados dos depósitos de quaisquer duas moedas sejam iguais quando medidos na mesma moeda é chamado de condição de paridade de juros. Isso implica que os detentores potenciais de depósitos em moeda estrangeira vêem todos eles como ativos igualmente desejáveis, desde que suas taxas esperadas de retorno sejam as mesmas. Dado que o retorno esperado sobre os depósitos em dólares dos EUA é 4% maior que o dos depósitos cedi no Gana, todas as coisas sendo iguais, ninguém estará disposto a continuar com os depósitos cedi do Gana e os detentores de depósitos cedi no Gana estarão tentando vendê-los Para depósitos em dólares. Haverá, portanto, um excesso de oferta de depósitos de cedi no Gana e uma demanda excessiva de depósitos em dólares dos EUA no mercado de câmbio (Krugman et al., 2012). Uma teoria importante da relação entre taxa de inflação e taxa de juros é o efeito Fisher às vezes referido como a hipótese de Fisher por Irvin Fisher. Fisher provou matematicamente que a taxa de juros nominal é igual à taxa de juros real menos a taxa de inflação esperada (prevista). O efeito Fisher simplesmente explica, por exemplo, que se a taxa de juros nominal for de 50% para um determinado período, ea taxa de inflação prevista nesse mesmo período for de 20%, a taxa de juros real será de 30%. The movement in short term interest rates primarily reflects fluctuation in expected inflation, which in effect has a predictive ability for future inflation (Mishkin and Simon 1995 ). The primary objective of the Central Bank of Ghana is to maintain stability in the general level of prices (Bank of Ghana Act 2002 ). Price Stability is, therefore, one of the most important indicators of the health of a nations economy. It must be noted that price stability alone might not be enough for a healthy economy. Several studies have been conducted on modelling inflation rates in Ghana, and majority of these used the constant variance assumption model. Although Mbeah-Baiden ( 2013 ) used non-constant variance models to model inflation rates in Ghana, his work only considered a univariate analysis of inflation rates. In the developed countries where a number of the researchers have modelled financial data series using Multivariate Generalized Autoregressive Conditional Heteroscedastic (MGARCH) models, none has modelled the co-movements of inflation rates, exchange rates and interest rates. The MGARCH models have not been explored enough on Ghanaian data and to a very large extent, Africa. It must be noted that Atta-Mensah and Bawumia ( 2003 ) used Vector Error Correction forecasting model for Ghana and concluded that growth rate, broad money supply (M2) and depreciation of exchange rate are the main drivers of higher inflation. The main objective of the study is to investigate the volatility and conditional relationship of inflation, exchange and interest rates and to construct a model using the multivariate GARCH BEKK (Baba, Engle, Kraft and Kroner) and DCC (Dynamic Conditional Correlation) models. A researcher can apply all these models on data series and the best model is chosen based on the performance of the model using a criterion. According to (Doan: RATS Handbook for ARCH/GARCH and Volatility Models. pp: 38. Evanston, United States: Estima, Unpublished Draft Book), the application of BEKK and DCC in modelling the conditional variance generally achieved similar results and the difference is negligible. Data and methodology The monthly inflation rates, average monthly exchange rates (cedi to US dollar) and interest rates (lending rate to the public) in Ghana spanning the period January 1990 to December 2013 were used for the study. This means that a total of 288 data points were considered for each variable. The sources of data were the Ghana Statistical Service (GSS) and Ghana Commercial Bank (GCB). The data were analyzed using multivariate GARCH, DCC and BEKK models. The procedure most often used in the model estimation involves the maximization of a likelihood function constructed on the assumption of independently and identically distributed standardized residuals. According to Engle and Sheppard ( 2001 ), analyzing and understanding how the univariate GARCH works is fundamental for the study of the Dynamic Conditional Correlation multivariate GARCH model. The DCC model is a nonlinear combination of univariate GARCH and its matrix is based on how the univariate GARCH (1, 1) process works. Suppose that the stochastic process ( trighttT ) denotes the return during a specific time period, where x t is the return observed at time t . Assuming for instance that the model for a return is given as: x t t t . where t ( x t / t 1 ) denotes the conditional expectation of the return series, t is the condition error and t 1 ( x . s t 1) represent the sigma field (information set) generated by the values of the return until time t - 1. Suppose that the conditional errors are conditional standard deviations of the returns ( t Var t/ right) ) times is independent and identically normally distributed with zero mean and a unit variance stochastic variable y t . Note that h t and y t are independent for all time t . ( tsqrt tsim Nleft(0, tright) ). Lastly, assume that the conditional expectation t 0, which implies that ( tsqrt t ) and x t / t 1 N (0, h t ). Conditioning of economic and financial models are mostly stated as the regression of a variables present values of the variable on the same variables past values as indicated in the GARCH(p, q) model proposed by (Bolleslev 1986 ) is given in equation ( 1 ): begin tphi sum p i 2sum q i kern1em, kern1em pge 0kern1.12em, kern1em qgt0 phi ge 0kern0.5em, kern0.5em ige 0,kern1em i1,2,3,dots, p ige 0kern1em forkern1em i1,2,3. . q. end The GARCH(p, q) consist of the three terms, these are: (i) The estimated DCC models unconditional covariance matrix is given in equation ( 12 ): Figure 3 displays the conditional correlation between inflation rates and exchange rates from 1990 to 2013. The plot indicates that there is a positive conditional association between inflation and exchange rate. This implies that, as the local currency the cedi depreciates to the US dollar, general levels of prices in Ghana also increases. The relationship was relatively stronger in 1991 and 1993 compared to 1992, the year election was held. The period of 1995, 1996 and 1997 as well as the years between 2003 and 2009 exhibited relatively weak correlation. Contrary, the period between 2000 and 2002 exhibited the strongest positive relationship. Depreciation of the cedi means that the cedi buys less than the US dollar, therefore, imports are more expensive and exports are cheaper. The positive relationship in the exchange rate depreciation and inflation rate means that, imported goods and services become more expensive and this affects the health of the economy especially because Ghana depends heavily on imported goods. The relationship exhibited is disincentive to cutting cost for companies whose raw materials are imported, this implies that depreciation causes cost-push inflation in the long run. Time series plot of the conditional correlation of inflation and exchange rates from 1990 to 2013. Table 2 displays seven months out-of-sample forecast of inflation rate for 2014 using the mean equation of the DCC model. The forecasts, compared to the observed rates declared by the Ghana Statistical Service indicate that there is evidence that the mean equation of the DCC model is robust in predicting inflation rate in the medium to short term. The widening of the error with time is an indication that general prices of goods and services react to the depreciation of the cedi or volatility in the exchange rate in the long run. Based on the DCC model, the mean equation is given as Seven months of 2014 out-sample forecast of inflation rate from the mean equation of the DCC model BEKK model The parameters A, B and C in the BEKK model are provided below: begin mathrmleft(begin hfill 0.40795405hfill amp hfill 0.04365957hfill amp hfill -0.0071803hfill hfill 0.13445156hfill amp hfill 0.93021649hfill amp hfill 0.07466339hfill hfill -0.9793208hfill amp hfill -0.2015772hfill amp hfill 0.01600388hfill end right),mathrm left(begin hfill 0.03700521hfill amp hfill 0.02687299hfill amp hfill 0.16010347hfill hfill -0.0394133hfill amp hfill -0.022465hfill amp hfill 0.13606053hfill hfill -0.003041hfill amp hfill 0.00067659hfill amp hfill -0.0419268hfill end right), mathrm left(begin hfill 7.85009270hfill amp hfill hfill amp hfill hfill hfill -0.0358002hfill amp hfill 1.59068051hfill amp hfill hfill hfill 0.03869339hfill amp hfill 0.20402223hfill amp hfill 2.94049023hfill end right)end Figure 4 exhibits the time series forecast of volatility in inflation, exchange and interest rates for the next twelve months. The exchange rates forecast indicates that there is likely to be instability in the exchange rate in 2014. This implies that the cedi is likely to deviate abnormally in 2014, that is, the cedi is expected to depreciate very fast in 2014. The inflation rate forecast suggest that, in 2014, general prices of goods and services will increase but at a low rate, interest rates will also increase at the same pace. The forecasts suggest economic instability in Ghana in 2014. The shocks in the graph suggest that inflation and interest rates react to exchange rates volatility in the medium to long term. As at the time of completing this research work, the cedi has depreciated 31.8 on June 5, 2014, per information available on the Bank of Ghana website, a record high within the last decade, (Bank of Ghana, 2014 ). The current rate of 31.8 suggest that inflation rates could escalate further if the cedi is not stabilized by the last quarter of 2014. Time series forecasts of volatility in inflation, exchange and interest rates. Certainly, it is evident that the BEKK model is robust in modeling volatility in the depreciation of the cedi to other foreign currencies. Figure 5 displays time series plot of inflation rates volatility from 1990 to 2013. There is evidence of relatively mild volatility in 2004 and 2008. Volatility in inflation rate during the study period could be found in 1993, 1995, 2003, 2004, 2005, 2007, 2008, 2010, 2011 and 2012. It must be noted that, the highest shock was in 2002. The risk in inflation means that there is evidence of abrupt deviation from the mean of the general level of prices of goods and services. The volatility exhibited during these periods implies that the expected inflation deviated from the observed mean value. Inflation volatility measures the uncertainty in the expected inflation. Volatility of any kind is likely to deteriorate the prospects of a healthy economy, if volatility is high investors become uncertain in their future investments since there is a high inflation risk, therefore demand a high return. High volatility in inflation leads to high cost of borrowing which directly affect investment negatively and to a large extent the health of the economy leading to ineffective planning. The trend in the plots indicates that inflation volatility trail exchange rate volatility this suggests that, inflation reacts to exchange rate volatility in the long run. Time series plot of inflation rate volatility from 1990 to 2013. Figure 6 is a time series plot of exchange rate volatility from 1990 to 2013. The period between 2002 and 2012 exhibited relatively mild deviation in mean exchange rate suggesting stability. Much of the turbulence could be observed between 2001 and 1990 as well as in 2013. The plot seems to suggest that exchange rate exhibits some sort of shocks a year after the general presidential and parliamentary elections are held in Ghana. It also suggests that the cedi depreciates fast during the first quarter of every year. The shocks in exchange rate impacts negatively on the economy of Ghana since it weakens the Ghanaian cedi against the US dollar. Volatility in the exchange rate will result in high prices of imported goods and services and reduces investor confidence in the economy. This implies that there will be uncertainty in the expectation of how the cedi will perform on the forex, as such many are likely to speculate, the public react by demanding more dollars, all things being equal, the cedi will depreciate further. The gross domestic product, employment and the overall health of the economy of Ghana will be affected negatively as a result. Time series plot of exchange rate volatility from 1990 to 2013. Vector error correction model and granger causality The Vector Error Correction Model and Granger Causality test is used to examine the cause and effect of the inflation rate, exchange rate and interest rate. Johansen test of cointegration among the variables using STATA 12 rejected the null hypothesis that there is no cointegration a precondition to running the Vector Error Correction model as shown in Table 3. Johansen test of cointegration among the variables using STATA 12 The Vector Correction Model evidence long run and short run causality among the variables after the null hypothesis of both no long run causality and no short run causality were rejected. After a pair-wise Granger-causality tests at 5 significant level, the result show that, exchange rate Granger-cause inflation rate but the converse does not. Similarly, inflation rate Granger cause interest rate but the reverse does not. Conclusions Multivariate GARCH, DCC and BEKK models were fitted to the variances of the data. Both models passed the diagnostic test. The mean equation of the DCC model was used to predict the expected inflation rate and proved to be robust in the short to medium term, similarly, the BEKK model was used to predict the expected exchange rate volatility. These predictions suggest that, inflation rates are expected to increase at a very slow rate in 2014. Also, the forecast of exchange rate volatility suggested that, there is a very high risk of abrupt depreciation of the cedi to the US dollar. This implies that the rates of inflation as well as interest rates are likely to react in the long run to the expected volatility in exchange rate for the year 2014. There was generally positive conditional and unconditional correlation between inflation rates and exchange rates, inflation rates and interest rates as well as exchange rates and interest rates. This implies that there is some evidence that when the general prices of goods and services are stable, interest rates are expected to be stable and possibly low. That of inflation and exchange rates implies that the stability of inflation means that the cedi depreciated to the dollar at low rate. There was evidence that the cedi has depreciated cumulatively to the US dollar of 7010.02 from 1990 to 2013 with a weighted annual average depreciation of 20.4. The volatility experienced in inflation, exchange and interest rates in the study, to a large extent were not in elections year. It is therefore factually inaccurate to assert that during election years, the cedi depreciates faster to the US dollar. The evidence rather suggests, there seem to be volatility in these economic variables, periods after elections were held rather than during elections year and also during the first quarter of every year. It was also evident that, the fact that inflation rates were stable, does not mean that exchange rates and interest rates are expected to be stable. Rather, when the cedi performs well on the forex, inflation and interest rates react positively in the long run. All things being equal, this reaction tickles down to all aspects of the economy thus, occasioning improved standards of living. The economy of Ghana reacts positively in most instances when the cedi performs strongly on the forex market. Such performance was evidenced in 2003 when the cedi depreciated to the US dollar at an average of 3.81, during that same year the Ghana Stock Exchange recorded returns on investments of about 155, the highest since its inception. The success of the cedi during this year could be traced to foreign inflows of HIPC benefits into the country. This implies that the health of the economy of Ghana is highly dependent on the strength of the cedi against foreign currencies such as the US dollar, Euro and the British pound sterling. Recommendations Recommendations are made for both policy formulation and areas of further research based on the findings of the study. To begin with, it is recommended that policy makers use multivariate GARCH models to study the dynamics of economic and financial data. The DCC model proved to be robust in modeling the correlation among inflation, exchange and interest rates, and the mean equation of the model was robust for modelling inflation rates in the short to medium term. Similarly, the BEKK model was found to be robust in modeling volatility as well as forecasting. Secondly, the research work has revealed that, the health of Ghanas economy is highly dependent on the strength of the Ghanaian currency: cedi against the foreign currencies since the country is import dependent, as such there must be a national agenda to increase foreign inflows and introduce a policy aimed at Exchange Rate Targeting (ERT). The forecast is also an indication that policy makers and industry players can effectively plan to curb uncertainties in the Ghanaian economy given these models are used. Thirdly, there must be a national consensus to reduce imports into the country by improving production and in the long run increase non-traditional exports. The government could adopt a policy through consensus with private sectors (services) to list on the Ghana Stock Exchange to attract Ghanaians to own shares, tax incentives could be used as a stimulus package. This is to ensure that 100 of the profit is not repatriated. Government could also dialogue with the private sector and propose a policy that mandates foreign owned companies to delay about 50 repatriation of their profit in the economy of Ghana for about two years. Government must also adopt a policy to reduce the number of State delegations to international events abroad to about 20, this could also reduce the pressure on the Ghanaian cedi. Lastly, a study into the dynamics of interest rates, stock returns and exchange rates is recommended. Other economic indicators such as money supply, balance of payment and budget deficit could be added to inflation rate, exchange rate and interest for modelling using multivariate GARCH models. Modelling the volatility in the five most traded currencies in Ghana is also recommended. Impulse analysis of inflation rates, exchange rates and interest rates is suggested as well. Declarations Competing interests We certify that there is no conflict of interest with any organization regarding the material and the research discussed in the manuscript. This work is also not financed by any entity. Authors contributions ENNN drafted the theoretical framework, methodology and literature review. DN did some literature review, helped with the analysis and some of the write-up. KD-A helped with the theoretical underpinning of the methodology as well as the discussions. KO-B helped with the economic review of theories and economic explanations to the analysis. All authors read and approved the final manuscript. Authors AffiliationsThis paper was aimed at investigating the volatility and conditional relationship among inflation rates, exchange rates and interest rates as well as to construct a model using multivariate GARCH DCC and BEKK models using Ghana data from January 1990 to December 2013. The study revealed that the cumulative depreciation of the cedi to the US dollar from 1990 to 2013 is 7,010.2 and the yearly weighted depreciation of the cedi to the US dollar for the period is 20.4. There was evidence that, the fact that inflation rate was stable, does not mean that exchange rates and interest rates are expected to be stable. Rather, when the cedi performs well on the forex, inflation rates and interest rates react positively and become stable in the long run. The BEKK model is robust to modelling and forecasting volatility of inflation rates, exchange rates and interest rates. The DCC model is robust to model the conditional and unconditional correlation among inflation rates, exchange rates and interest rates. The BEKK model, which forecasted high exchange rate volatility for the year 2014, is very robust for modelling the exchange rates in Ghana. The mean equation of the DCC model is also robust to forecast inflation rates in Ghana. Modeling inflation rates and exchange rates in Ghana: application of multivariate GARCH models. Citations BioEntities Related Articles External Links Springerplus. 2015 4: 86. Published online 2015 February 24. doi: 10.1186/s40064-015-0837-6 Modeling inflation rates and exchange rates in Ghana: application of multivariate GARCH models Department of Statistics, University of Ghana, P. O. Box LG 115, Legon-Accra, Ghana Ghana Institute of Management and Public Administration Business School, Achimota-Accra, Ghana Received 2014 September 25 Accepted 2015 January 19. Copyright x000a9 Nortey et al. licensee Springer. 2015 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (creativecommons. org/licenses/by/4.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. This paper was aimed at investigating the volatility and conditional relationship among inflation rates, exchange rates and interest rates as well as to construct a model using multivariate GARCH DCC and BEKK models using Ghana data from January 1990 to December 2013. The study revealed that the cumulative depreciation of the cedi to the US dollar from 1990 to 2013 is 7,010.2 and the yearly weighted depreciation of the cedi to the US dollar for the period is 20.4. There was evidence that, the fact that inflation rate was stable, does not mean that exchange rates and interest rates are expected to be stable. Rather, when the cedi performs well on the forex, inflation rates and interest rates react positively and become stable in the long run. The BEKK model is robust to modelling and forecasting volatility of inflation rates, exchange rates and interest rates. The DCC model is robust to model the conditional and unconditional correlation among inflation rates, exchange rates and interest rates. The BEKK model, which forecasted high exchange rate volatility for the year 2014, is very robust for modelling the exchange rates in Ghana. The mean equation of the DCC model is also robust to forecast inflation rates in Ghana. Keywords: DCC, BEKK, GARCH, Ghana, Volatility, Inflation, Exchange, Interest rates Introduction When the general level of prices is relatively stable, the uncertainties of time-related activities such as investment diminish. This helps to promote full employment and strong economic growth. When price stability is achieved and maintained, monetary policy makers have done their job well (Sobel et al. 2006 ). Conceivably, one of the most important responsibilities of every government is fostering a healthy economy, which benefits all her citizens. The government through its ability to tax, spend and control money supply, attempts to promote full employment, price stability and economic growth. The importance of price stability is also emphasized in the Maastricht agreement, which defined the framework for a single European Currency, Euro, and identified price stability as the main objective of the new European Central Bank (McEachern 2006 ). Deflation could result to doom for an economy that is, it weakens consumer demand for goods and services as households are likely not to spend, believing that prices will continue to fall. This means that businesses as well as government may be unable to pay debts and could result in retrenchment. Emphasizing this point is Lagarde the Managing Director of the IMF, in April 2014 who cautioned the euro area that, a prolonged period of x0201clow-inflationx0201d or deflation can suppress demand and output, and overturn growth and jobs. According to Goldberg and Knetter (1997 ) exchange rate pass-through is the percentage change in local currency import prices resulting from a one percent change in the exchange rate between the exporting and importing countries. Exchange rate pass-through therefore is the effect (positive or negative) of exchange rates on import and export prices, consumer prices or inflation, investments as well as trade volumes. Engel and Rogers (1996 ) established that crossing the US-Canada border can considerably raise relative price volatility and that exchange rate fluctuations explain about one-third of the volatility increase. That is US-Canada border is an important determinant of relative price volatility even after making due allowance for the role of distance. Parsley and Wei ( 2001 ) confirmed previous findings that crossing national borders adds significantly to price dispersion. The demand for and supply of money are the key determinants of exchange rates. Interest Rate Parity is an important concept that explains the equilibrium state of the relationship between interest rate and exchange rate of two countries. The foreign exchange market is in equilibrium when deposits of all currencies offer the same expected rate of return. The condition that the expected returns on deposits of any two currencies are equal when measured in the same currency is called the interest parity condition. It implies that potential holders of foreign currency deposits view them all as equally desirable assets, provided their expected rates of return are the same. Given that the expected return on say US dollar deposits is 4 percent greater than that on Ghana cedi deposits, all things being equal, no one will be willing to continue holding Ghana cedi deposits, and holders of Ghana cedi deposits will be trying to sell them for US dollar deposits. There will therefore be an excess supply of Ghana cedi deposits and an excess demand for US dollar deposits in the foreign exchange market (Krugman et al. 2012 ). An important theory of the relationship between inflation rate and interest rate is the Fisher effect sometimes referred to as the Fisher hypothesis by Irvin Fisher. Fisher proved mathematically that the nominal interest rate is equal to the real interest rate minus the expected (predicted) inflation rate. The Fisher effect simply explains for example that if the nominal interest rate is say 50 per cent for a given period, and the predicted inflation rate during that same period is 20 per cent, then the real interest rate is 30 per cent. The movement in short term interest rates primarily reflects fluctuation in expected inflation, which in effect has a predictive ability for future inflation (Mishkin and Simon 1995 ). The primary objective of the Central Bank of Ghana is to maintain stability in the general level of prices (Bank of Ghana Act 2002 ). Price Stability is, therefore, one of the most important indicators of the health of a nationx02019s economy. It must be noted that price stability alone might not be enough for a healthy economy. Several studies have been conducted on modelling inflation rates in Ghana, and majority of these used the constant variance assumption model. Although Mbeah-Baiden (2013 ) used non-constant variance models to model inflation rates in Ghana, his work only considered a univariate analysis of inflation rates. In the developed countries where a number of the researchers have modelled financial data series using Multivariate Generalized Autoregressive Conditional Heteroscedastic (MGARCH) models, none has modelled the co-movements of inflation rates, exchange rates and interest rates. The MGARCH models have not been explored enough on Ghanaian data and to a very large extent, Africa. It must be noted that Atta-Mensah and Bawumia (2003 ) used Vector Error Correction forecasting model for Ghana and concluded that growth rate, broad money supply (M2) and depreciation of exchange rate are the main drivers of higher inflation. The main objective of the study is to investigate the volatility and conditional relationship of inflation, exchange and interest rates and to construct a model using the multivariate GARCH BEKK (Baba, Engle, Kraft and Kroner) and DCC (Dynamic Conditional Correlation) models. A researcher can apply all these models on data series and the best model is chosen based on the performance of the model using a criterion. According to (Doan: RATS Handbook for ARCH/GARCH and Volatility Models. pp: 38. Evanston, United States: Estima, Unpublished Draft Book), the application of BEKK and DCC in modelling the conditional variance generally achieved similar results and the difference is negligible. Data and methodology The monthly inflation rates, average monthly exchange rates (cedi to US dollar) and interest rates (lending rate to the public) in Ghana spanning the period January 1990 to December 2013 were used for the study. This means that a total of 288 data points were considered for each variable. The sources of data were the Ghana Statistical Service (GSS) and Ghana Commercial Bank (GCB). The data were analyzed using multivariate GARCH, DCC and BEKK models. The procedure most often used in the model estimation involves the maximization of a likelihood function constructed on the assumption of independently and identically distributed standardized residuals. According to Engle and Sheppard (2001 ), analyzing and understanding how the univariate GARCH works is fundamental for the study of the Dynamic Conditional Correlation multivariate GARCH model. The DCC model is a nonlinear combination of univariate GARCH and its matrix is based on how the univariate GARCH (1, 1) process works. Suppose that the stochastic process x t t T denotes the return during a specific time period, where x t is the return observed at time t . Assuming for instance that the model for a return is given as: x t x003bc t x003b5 t . where x003bc t x00395 ( x t / x003bb t x02212 1 ) denotes the conditional expectation of the return series, x003b5 t is the condition error and x003bb t x02212 1 x003c3 ( x : s x02264 t x02212 1) represent the sigma field (information set) generated by the values of the return until time t - 1. Suppose that the conditional errors are conditional standard deviations of the returns h t 1 / 2 V a r x t / x003bb t x02212 1 1 / 2 times is independent and identically normally distributed with zero mean and a unit variance stochastic variable y t . Note that h t and y t are independent for all time t . x003b5 t h t y t x0223c N 0. h t. Lastly, assume that the conditional expectation x003bc t 0, which implies that x t h t y t and x t / x003bb t x02212 1 Conditioning of economic and financial models are mostly stated as the regression of a variablex02019s present values of the variable on the same variablex02019s past values as indicated in the GARCH(p, q) model proposed by (Bolleslev 1986 ) is given in equation (1 ): h t x003d5 x02211 i 1 p x003b1 i x003b5 t x02212 1 2 x02211 i 1 q x003b2 i h t x02212 1. p x02265 0. q x0003e 0 x003d5 x02265 0. x003b1 i x02265 0. i 1. 2. 3. x02026. p x003b2 i x02265 0 f o r i 1. 2. 3. q. The GARCH(p, q) consist of the three terms, these are: (i) x003d5 - the weighted long run variance (ii) x02211 i 1 p x003b1 i x003b5 t x02212 1 2 - the moving average term, which is the sum of the p previous lags of squared-innovations multiplied by the assigned weight x003b1 i for each lagged square innovation (iii) x02211 i 1 q x003b2 i h t x02212 1 - the autoregressive term, which is the sum of the q previous lagged variances multiplied by the assigned x003b2 i for each lagged variance. Since the variance is non-negative by definition, the process h t t 0 x0221e must also be non-negative valued. Baba, Engle, Kraft and Kroner (BEKK) model To ensure positive definiteness, a new parameterization of the conditional variance matrix H t was defined by, (Baba, Engle, Kraft, Kroner: Multivariate simultaneous generalized ARCH at the University of California, San Diego, unpublished) and became known as the BEKK model, which is viewed as another restricted version of the VEC model. It achieves the positive definiteness of the conditional covariance by formulating the model in a way that this property is implied by the model structure. The form of the BEKK model is as: H t C C x02211 j 1 q x02211 k 1 K A k j x003b5 t x02212 j x003b5 t x02212 j A k j x02211 j 1 p x02211 k 1 K B k j H t x02212 j B k j where A kj and B kj a N x000d7 N parameter matrices, and C is a lower triangular matrix. The purpose of decomposing the constant term in equation (2 ) into a product of the two triangular matrices is to guarantee the positive semi-definiteness of H t . Whenever K x0003e 1 an identification problem would be generated for the reason that there are not only a single parameterization that can obtain the same representation of the model. The first-order BEKK model is given as: H t C C A x003b5 t x02212 j x003b5 t x02212 j A B H t x02212 j B The BEKK model specified in equation (3 ) also has its diagonal form by assuming that the matrices A kj and B kj are diagonal. It is a restrictive version of the DVEC model. The most restricted version of the diagonal BEKK is the scalar BEKK one with A aI and B bI where x003b1 and b are scalars. Estimation of the BEKK model still bears large computations due to several matrix transpositions. The number of parameters of a complete BEKK model is ( p q ) KN 2 N ( N 1)/2 whereas in the diagonal BEKK, the number of parameters reduces to ( p q ) KN N ( N 1)/2. The BEKK form is not linear in the parameters, which makes the convergence of the model difficult. However, the model structure automatically guarantees the positive definiteness of H t . Under the overall consideration, it is assumed that p q k 1 in BEKK forms of application. The difference between the results of BEKK model and the DCC model is highly negligible. The Dynamic Conditional Correlation (DCC) Model To extend the assumptions in the univariate GARCH to multivariate case, suppose that we have n assets in a portfolio and the return vector is x t ( x 1 t , x 2 t , x 3 t , x02026, x nt ) . Furthermore, assume that the conditional returns are normally distributed with zero mean and conditional covariance matrix H t E x t x t. / x003bb t x02212 1. This implies that x t H t 1 / 2 y t and x t / x003bb t x02212 1 N (0, I n ) and I n is the identity matrix of order n. H t 1 / 2 may be obtained by Cholesky decomposition of H t . In DCC-model, the covariance matrix is decomposed into H t x02261 D t X t D t . where D t is the diagonal matrix of time varying standard variation from univariate GARCH process D t h 1 t 0 x022ef 0 0 h 2 t x022ef 0 x022ee x022ee x022f1 x022ee 0 0 x022ef h nt. The specification of elements in the D t matrix is not only restricted to the GARCH(p, q) described in equation (1 ) but to any GARCH process with normally distributed errors which meet the requirements for suitable stationary and non-negative conditions. The number of lags for each assets and series do not need to be the same either. However, X t is the conditional correlation matrix of the standardized disturbances x003b5 t where X t 1 Q 12. t x022ef Q 1 n. t Q 21. t 1 x022ef Q 2 n. t x022ee x022ee x022f1 x022ee Q n 1. t Q n 2. t x022ef 1 and x003b5 t D t x02212 1 x t x0223c N 0. X t. Thus, the conditional correlation is the conditional covariance between the standardized disturbances. By the definition of the covariance matrix, H t has to be positive definite. Since H t is a quadratic form based on X ts . it follows from basics in linear algebra that X t has to be positive definite to ensure that H t is positive definite. By the definition of the conditional correlation matrix all the elements have to be equal to or less than one. To ensure that all of these requirements are met, X t is decomposed into X t Q t x02212 1 Q t Q t x02212 1. where Q t is a positive definite matrix defining the structure of the dynamics and Q t x02212 1 rescales the elements in Q t to ensure that q ij x02264 1. This implies that, Q t x02212 1 is simply the inverted diagonal matrix with the squared root diagonal elements of Q t . Q t x02212 1 1 / Q 11 t 0 x022ef 0 0 1 / Q 11 t x022ef 0 x022ee x022ee x022f1 0 0 0 x022ef 1 / Q 11 t. Suppose that Q t has the following dynamics: Q t 1 x02212 x003b1 x02212 x003b2 Q x000af x003b1 x003b5 t x02212 1 x003b5 t x02212 1. x003b2 Q t x02212 1 where Q x000af is the unconditional covariance of the standardized disturbances Q x000af cov x003b5 t x003b5 t. E x003b5 t x003b5 t. and x003b2 are scalars. The dynamic structure defined above is the simplest multivariate GARCH called Scalar GARCH. A major caveat of this structure is the all correlations obey the same structure. The structure can be extended to the general DCC(P, Q) Q t 1 x02212 x003a3 i 1 P x003b1 i x02212 x003a3 j 1 Q x003b2 j Q x000af x003a3 i 1 P x003b1 i x003b5 t x02212 1 x003b5 t x02212 1. x003a3 j 1 Q x003b2 j Q t x02212 1 In this work, only the DCC(1,1) will be utilized. Constraints of the DCC(1,1) Model If the covariance matrix is not positive definite then it is impossible to invert the covariance matrix which is essential in portfolio optimization. To guarantee a positive definite H t for all t . simple conditions on the parameters are imposed. Firstly, the conditions for the univariate GARCH model has to be satisfied. Similar conditions on the dynamic correlations are also required, namely: x003b2 x02265 0 and x003b1 x02265 0, x003b1 x003b2 x0003c 1, Q 0 has to be positive definite. Estimation of the DCC(1,1) Model In order to estimate the parameters of H t . that is to say x003b8 ( x003b8 1 , x003b8 2 ), the following log-likelihood function can be used when the errors are assumed to be multivariate normally distributed: x02113 x003b8 x02212 1 2 x02211 t 1 T n log 2 x003c0 log H t x t H t x02212 1 x t x02113 x003b8 x02212 1 2 x02211 t 1 T n log 2 x003c0 log D t X t D t x t D t x02212 1 X t x02212 1 D t x02212 1 x t x02113 x003b8 x02212 1 2 x02211 t 1 T n log 2 x003c0 2 log D t log X t x003b5 t X t x02212 1 x003b5 t The parameters in the DCC(1,1) model specified in equation (6 ) can be divided into two groups, that is: x003b8 1 x003d5 1. x003b1 1. x003b2 1. x003d5 2. x003b1 2. x003b2 2 x02026. x003d5 n. x003b1 n. x003b2 n and x003b8 2 x003b7. x003c8. The estimation follows the following two steps. The X t matrix in the log-likelihood function is replaced with the identity matrix I n . which gives the following log-likelihood function specified in equation (8 ): x02113 x003b8 1 / x t x02212 1 2 x02211 t 1 T n log 2 x003c0 2 log D t log I n x t D t x02212 1 I n D t x02212 1 x t x02113 x003b8 1 / x t x02212 1 2 x02211 t 1 T n log 2 x003c0 2 log D t x t D t x02212 1 D t x02212 1 x t x02113 x003b8 1 / x t x02212 1 2 x02211 t 1 T x02211 i 1 n log 2 x003c0 log h it r it 2 h it x02113 x003b8 1 / x t x02212 1 2 x02211 i 1 n T log 2 x003c0 x02211 t 1 T log h it r it 2 h it It is obvious that this quasi-likelihood function is the sum of the univariate GARCH log-likelihood functions. Therefore, one can use the algorithm to estimate parameter x003b8 1 ( x003d5 1 , x003b1 1 , x003b2 1 , x003d5 2 , x003b1 2 , x003b2 2 x02026, x003d5 n , x003b1 n , x003b2 n ) for each univariate GARCH process. Since the variance h it for asset i 1, 2, 3, x02026 n is estimated for t T , then also the element of the D t matrix under the same time period is estimated. In the second step, the correctly specified log-likelihood function is used to estimate x003b8 2 ( x003b7 , x003c8 ) given the estimated parameters x003b8 1 x003d5 1. x003b1 1. x003b2 1. x003d5 2. x003b1 2. x003b2 2 x02026. x003d5 n. x003b1 n. x003b2 n from step one, we obtain: x02113 2 x003b8 2 / x003b8 1. x t x02212 1 2 x02211 t 1 T n log 2 x003c0 2 log D t log X t x003b5 t X t x02212 1 x003b5 t From equation (9 ), the first two terms in the log-likelihood are constants therefore, the two last terms including X t is of interest to maximize. Hence we obtain: x02113 2 x0221d log X t x003b5 t X t x02212 1 x003b5 t. Q x000af is estimated as: Q 1 T x02211 t 1 T x003b5 t x003b5 t . Variance targeting is used in the dynamic structure and therefore Q 0 x003b5 0 x003b5 0 and since the conditional correlation matrix also is the covariance matrix of the standardized residuals, X 0 x003b5 0 x003b5 0 . Results and discussion Figurex000a0 1 shows the time series plot for inflation rates, exchange rates and interest rates from 1990 to 2013 based on R output. The inflation rates and interest rates plots exhibits downward trend with fluctuations, contrarily, the exchange rates plot exhibit continuous upwards trend. The movements of the plots indicate that the mean and the variance of the exchange rates data are changing overtime. This means that the mean is non constant and the variance is unstable. Time series plot of inflation, exchange and interest rates from 1990 to 2013. Figurex000a0 2 displays the time series plot of the natural logarithm of inflation rates, exchange rates and interest rates from January 1990 to December 2013 using RATS 8.3. The time series plot appears to be stable after the transformation using the natural logarithm of inflation, exchange and interest rates. This suggests that the mean and variance are stable over time implying that the variables achieve stationarity after taking the natural logarithm. Time series plot of natural logarithm of monthly inflation rate exchange rate and interest rate in Ghana from 1990 to 2013. The cumulative depreciation of the cedi to the US dollar from 1990 to 2013 is 7,010.2 and the yearly weighted depreciation of the cedi to the US dollar is 20.4 using the formulae in equation (11 ) and (12) respectively Depreciation rat e end x02212 rat e initial rat e initial x000d7 100. where n is the number of years. Multivariate-GARCH modeling Multivariate GARCH models are estimated by the quasi maximum likelihood technique. Regression Analysis of Time Series (RATS) 8.3 is a widely used software in estimating MGARCH models as a result of its flexible maximum likelihood estimation capabilities and has advantages over many other software packages on estimating MGARCH models. The optimization algorithm used for the maximum likelihood estimation in RATS is BFGS proposed independently by Broyden (1970 ), Fletcher (1970 ), Goldfarb (1970 ) and Shanno (1970 ), (Estima 2013 ). This optimization algorithm uses iteration routines to obtain the coefficient estimation. As such, convergence is assumed to occur if the change in the coefficient to be estimated is less than the criterion option 0.00001 specified. RATS was used in estimating the MGARCH models for this study. Tablex000a0 1 shows both the DCC and BEKK with respective p-values of 0.99659 and 0.9869. The p-values are greater than a significance level of 0.05, hence it can be concluded that the there is no multivariate ARCH effect. This also suggests that the conditional distribution of the white noise is near Gaussian. Test of multivariate ARCH effect and serial correlation of DCC and BEKK The estimated DCC modelx02019s unconditional covariance matrix is given in equation (12 ): h 11 t 48.7058399 0.2821624 x003bc 1. t x02010 1 2 0.0410249 h 11 t x02010 1 h 22 t 1.5122533 0.23933668 x003bc 2. t x02010 1 2 0.48564696 h 22 t x02010 1 h 33 t 3.82780228 0.0107313 x003bc 3. t x02010 1 2 0.70807305 h 33 t x02010 1 Q t 1 x02212 0.01007687 x02212 0.9705411 Q x000af 0.01007687 x003b5 t x02010 1 x003b5 x02019 t x02010 1 0.9705411 Q t x02212 1 Q x000af 1.00000 x02212 0.03357 0.03980 x02212 0.03357 1.00000 x02212 0.86917 0.03980 x02212 0.86917 1.00000 Figurex000a0 3 displays the conditional correlation between inflation rates and exchange rates from 1990 to 2013. The plot indicates that there is a positive conditional association between inflation and exchange rate. This implies that, as the local currency the cedi depreciates to the US dollar, general levels of prices in Ghana also increases. The relationship was relatively stronger in 1991 and 1993 compared to 1992, the year election was held. The period of 1995, 1996 and 1997 as well as the years between 2003 and 2009 exhibited relatively weak correlation. Contrary, the period between 2000 and 2002 exhibited the strongest positive relationship. Depreciation of the cedi means that the cedi buys less than the US dollar, therefore, imports are more expensive and exports are cheaper. The positive relationship in the exchange rate depreciation and inflation rate means that, imported goods and services become more expensive and this affects the health of the economy especially because Ghana depends heavily on imported goods. The relationship exhibited is disincentive to cutting cost for companies whose raw materials are imported, this implies that depreciation causes cost-push inflation in the long run. Time series plot of the conditional correlation of inflation and exchange rates from 1990 to 2013. Tablex000a0 2 displays seven months out-of-sample forecast of inflation rate for 2014 using the mean equation of the DCC model. The forecasts, compared to the observed rates declared by the Ghana Statistical Service indicate that there is evidence that the mean equation of the DCC model is robust in predicting inflation rate in the medium to short term. The widening of the error with time is an indication that general prices of goods and services react to the depreciation of the cedi or volatility in the exchange rate in the long run. Based on the DCC model, the mean equation is given as Inflatio n t 48.7058399 x02212 0.12070302 t BEKK model The parameters A, B and C in the BEKK model are provided below: A 0.40795405 0.04365957 x02212 0.0071803 0.13445156 0.93021649 0.07466339 x02212 0.9793208 x02212 0.2015772 0.01600388. B 0.03700521 0.02687299 0.16010347 x02212 0.0394133 x02212 0.022465 0.13606053 x02212 0.003041 0.00067659 x02212 0.0419268. C 7.85009270 x02212 0.0358002 1.59068051 0.03869339 0.20402223 2.94049023 Figurex000a0 4 exhibits the time series forecast of volatility in inflation, exchange and interest rates for the next twelve months. The exchange rates forecast indicates that there is likely to be instability in the exchange rate in 2014. This implies that the cedi is likely to deviate abnormally in 2014, that is, the cedi is expected to depreciate very fast in 2014. The inflation rate forecast suggest that, in 2014, general prices of goods and services will increase but at a low rate, interest rates will also increase at the same pace. The forecasts suggest economic instability in Ghana in 2014. The shocks in the graph suggest that inflation and interest rates react to exchange rates volatility in the medium to long term. As at the time of completing this research work, the cedi has depreciated 31.8 on June 5, 2014, per information available on the Bank of Ghana website, a record high within the last decade, (Bank of Ghana, 2014 ). The current rate of 31.8 suggest that inflation rates could escalate further if the cedi is not stabilized by the last quarter of 2014. Time series forecasts of volatility in inflation, exchange and interest rates. Certainly, it is evident that the BEKK model is robust in modeling volatility in the depreciation of the cedi to other foreign currencies. Figurex000a0 5 displays time series plot of inflation rates volatility from 1990 to 2013. There is evidence of relatively mild volatility in 2004 and 2008. Volatility in inflation rate during the study period could be found in 1993, 1995, 2003, 2004, 2005, 2007, 2008, 2010, 2011 and 2012. It must be noted that, the highest shock was in 2002. The risk in inflation means that there is evidence of abrupt deviation from the mean of the general level of prices of goods and services. The volatility exhibited during these periods implies that the expected inflation deviated from the observed mean value. Inflation volatility measures the uncertainty in the expected inflation. Volatility of any kind is likely to deteriorate the prospects of a healthy economy, if volatility is high investors become uncertain in their future investments since there is a high inflation risk, therefore demand a high return. High volatility in inflation leads to high cost of borrowing which directly affect investment negatively and to a large extent the health of the economy leading to ineffective planning. The trend in the plots indicates that inflation volatility trail exchange rate volatility this suggests that, inflation reacts to exchange rate volatility in the long run. Time series plot of inflation rate volatility from 1990 to 2013. Figurex000a0 6 is a time series plot of exchange rate volatility from 1990 to 2013. The period between 2002 and 2012 exhibited relatively mild deviation in mean exchange rate suggesting stability. Much of the turbulence could be observed between 2001 and 1990 as well as in 2013. The plot seems to suggest that exchange rate exhibits some sort of shocks a year after the general presidential and parliamentary elections are held in Ghana. It also suggests that the cedi depreciates fast during the first quarter of every year. The shocks in exchange rate impacts negatively on the economy of Ghana since it weakens the Ghanaian cedi against the US dollar. Volatility in the exchange rate will result in high prices of imported goods and services and reduces investor confidence in the economy. This implies that there will be uncertainty in the expectation of how the cedi will perform on the forex, as such many are likely to speculate, the public react by demanding more dollars, all things being equal, the cedi will depreciate further. The gross domestic product, employment and the overall health of the economy of Ghana will be affected negatively as a result. Time series plot of exchange rate volatility from 1990 to 2013. Vector error correction model and granger causality The Vector Error Correction Model and Granger Causality test is used to examine the cause and effect of the inflation rate, exchange rate and interest rate. Johansen test of cointegration among the variables using STATA 12 rejected the null hypothesis that there is no cointegration a precondition to running the Vector Error Correction model as shown in Tablex000a0 3 . Johansen test of cointegration among the variables using STATA 12 The Vector Correction Model evidence long run and short run causality among the variables after the null hypothesis of both x0201cno long run causality and no short run causalityx0201d were rejected. After a pair-wise Granger-causality tests at 5 significant level, the result show that, exchange rate Granger-cause inflation rate but the converse does not. Similarly, inflation rate Granger cause interest rate but the reverse does not. Conclusions Multivariate GARCH, DCC and BEKK models were fitted to the variances of the data. Both models passed the diagnostic test. The mean equation of the DCC model was used to predict the expected inflation rate and proved to be robust in the short to medium term, similarly, the BEKK model was used to predict the expected exchange rate volatility. These predictions suggest that, inflation rates are expected to increase at a very slow rate in 2014. Also, the forecast of exchange rate volatility suggested that, there is a very high risk of abrupt depreciation of the cedi to the US dollar. This implies that the rates of inflation as well as interest rates are likely to react in the long run to the expected volatility in exchange rate for the year 2014. There was generally positive conditional and unconditional correlation between inflation rates and exchange rates, inflation rates and interest rates as well as exchange rates and interest rates. This implies that there is some evidence that when the general prices of goods and services are stable, interest rates are expected to be stable and possibly low. That of inflation and exchange rates implies that the stability of inflation means that the cedi depreciated to the dollar at low rate. There was evidence that the cedi has depreciated cumulatively to the US dollar of 7010.02 from 1990 to 2013 with a weighted annual average depreciation of 20.4. The volatility experienced in inflation, exchange and interest rates in the study, to a large extent were not in elections year. It is therefore factually inaccurate to assert that during election years, the cedi depreciates faster to the US dollar. The evidence rather suggests, there seem to be volatility in these economic variables, periods after elections were held rather than during elections year and also during the first quarter of every year. It was also evident that, the fact that inflation rates were stable, does not mean that exchange rates and interest rates are expected to be stable. Rather, when the cedi performs well on the forex, inflation and interest rates react positively in the long run. All things being equal, this reaction tickles down to all aspects of the economy thus, occasioning improved standards of living. The economy of Ghana reacts positively in most instances when the cedi performs strongly on the forex market. Such performance was evidenced in 2003 when the cedi depreciated to the US dollar at an average of 3.81, during that same year the Ghana Stock Exchange recorded returns on investments of about 155, the highest since its inception. The success of the cedi during this year could be traced to foreign inflows of HIPC benefits into the country. This implies that the health of the economy of Ghana is highly dependent on the strength of the cedi against foreign currencies such as the US dollar, Euro and the British pound sterling. Recommendations Recommendations are made for both policy formulation and areas of further research based on the findings of the study. To begin with, it is recommended that policy makers use multivariate GARCH models to study the dynamics of economic and financial data. The DCC model proved to be robust in modeling the correlation among inflation, exchange and interest rates, and the mean equation of the model was robust for modelling inflation rates in the short to medium term. Similarly, the BEKK model was found to be robust in modeling volatility as well as forecasting. Secondly, the research work has revealed that, the health of Ghanax02019s economy is highly dependent on the strength of the Ghanaian currency: cedi against the foreign currencies since the country is import dependent, as such there must be a national agenda to increase foreign inflows and introduce a policy aimed at Exchange Rate Targeting (ERT). The forecast is also an indication that policy makers and industry players can effectively plan to curb uncertainties in the Ghanaian economy given these models are used. Thirdly, there must be a national consensus to reduce imports into the country by improving production and in the long run increase non-traditional exports. The government could adopt a policy through consensus with private sectors (services) to list on the Ghana Stock Exchange to attract Ghanaians to own shares, tax incentives could be used as a stimulus package. This is to ensure that 100 of the profit is not repatriated. Government could also dialogue with the private sector and propose a policy that mandates foreign owned companies to delay about 50 repatriation of their profit in the economy of Ghana for about two years. Government must also adopt a policy to reduce the number of State delegations to international events abroad to about 20, this could also reduce the pressure on the Ghanaian cedi. Lastly, a study into the dynamics of interest rates, stock returns and exchange rates is recommended. Other economic indicators such as money supply, balance of payment and budget deficit could be added to inflation rate, exchange rate and interest for modelling using multivariate GARCH models. Modelling the volatility in the five most traded currencies in Ghana is also recommended. Impulse analysis of inflation rates, exchange rates and interest rates is suggested as well. We certify that there is no conflict of interest with any organization regarding the material and the research discussed in the manuscript. This work is also not financed by any entity. ENNN drafted the theoretical framework, methodology and literature review. DN did some literature review, helped with the analysis and some of the write-up. KD-A helped with the theoretical underpinning of the methodology as well as the discussions. 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