Purchasing Power Parity (PPP) has been regarded as the most important theory to explain the exchange rate movement based on relative price levels of two countries. After 1973, more and more countries were taking the floating exchange rate system, and the real exchange is testing out to be a non-stationary time seriess. This would be some real factors to have an effect on the real exchange rate. In the article, We study how these possible factors change the real exchange rate and make use of Wu et.al (2008) and Lee (2010)¡¦s local projection to estimate the impulse responses under the non-stationary time series which has cointegration vectors, and then we compare the difference between the impulse response in conventional VAR and the impulse response in Local Projection. The emprical model we use is the smae one as in Zhou (1995) and Wang and Dunne (2003), and the rule of the data is the same as in Wang and Dunne (2003). Finally, we get the consistent conclusion with Wu et.al (2008), Zhou (1995) and Wang and Dunne (2003).

Jorda (2005) proposed the new method to estimate impulse response functions by local projection. The new method, local projection, can avoid the misspecification problem. That is, local projections are robust to misspecification of the data generating process (DGP). Wu, Lee, and Wang (2008) extended the Jorda¡¦s local projection from stationary time series I(0) to non-stationary time series I(1). It makes the local projection be a more generally applicative method for the Macroeconomic. In the article, I relax the cointegration vector which assumed to be known in the Wu, Lee, and Wang (2008) and Lee(2010). From the inference of Johansen (1995) I can get the property of super-consistent between £] and ˆ £] in the cointegration vector. I use the above condition and OLS to estimate impulse response functions, and in the asymptotic theorem, the cointegration vectors which assumed to be known or estimated by Johansen MLE are both get the consistent coefficients of impulse responses.

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Thesis (MSc (Mathematics))--University of Stellenbosch, 2009. / Wavelet decomposition techniques have grown over the last two decades into a powerful tool in signal analysis. Similarly, spline functions have enjoyed a sustained high popularity in the approximation of data. In this thesis, we study the cardinal B-spline wavelet construction procedure based on quasiinterpolation and local linear projection, before specialising to the cubic B-spline on a bounded interval. First, we present some fundamental results on cardinal B-splines, which are piecewise polynomials with uniformly spaced breakpoints at the dyadic points Z/2r, for r ∈ Z. We start our wavelet decomposition method with a quasi-interpolation operator Qm,r mapping, for every integer r, real-valued functions on R into Sr m where Sr m is the space of cardinal splines of order m, such that the polynomial reproduction property Qm,rp = p, p ∈ m−1, r ∈ Z is satisfied. We then give the explicit construction of Qm,r. We next introduce, in Chapter 3, a local linear projection operator sequence {Pm,r : r ∈ Z}, with Pm,r : Sr+1 m → Sr m , r ∈ Z, in terms of a Laurent polynomial m solution of minimally length which satisfies a certain Bezout identity based on the refinement mask symbol Am, which we give explicitly. With such a linear projection operator sequence, we define, in Chapter 4, the error space sequence Wr m = {f − Pm,rf : f ∈ Sr+1 m }. We then show by solving a certain Bezout identity that there exists a finitely supported function m ∈ S1 m such that, for every r ∈ Z, the integer shift sequence { m(2 · −j)} spans the linear space Wr m . According to our definition, we then call m the mth order cardinal B-spline wavelet. The wavelet decomposition algorithm based on the quasi-interpolation operator Qm,r, the local linear projection operator Pm,r, and the wavelet m, is then based on finite sequences, and is shown to possess, for a given signal f, the essential property of yielding relatively small wavelet coefficients in regions where the support interval of m(2r · −j) overlaps with a Cm-smooth region of f. Finally, in Chapter 5, we explicitly construct minimally supported cubic B-spline wavelets on a bounded interval [0, n]. We also develop a corresponding explicit decomposition algorithm for a signal f on a bounded interval. ii Throughout Chapters 2 to 5, numerical examples are provided to graphically illustrate the theoretical results.

Doctor of Philosophy / Department of Economics / Steven P. Cassou / This dissertation consists of three essays in asymmetric empirical macroeconomics. Making macroeconomic policies has become increasingly difficult because of intricate relationships among macroeconomic variables. In this dissertation, we apply state-of-the-art macroeconometric techniques to investigate asymmetric relationships between key macroeconomic aggregates. Our findings have important macroeconomic policy implications. An analogue to the Phillips curve shows a positive relationship between inflation and capacity utilization. Some recent empirical work has shown that this relationship has broken down when using data after the mid-1980s and several popular explanations for this changing relationship, including advancements in technology and globalization, were put forward as possible explanations. In the first essay, we empirically investigate this issue using several threshold error correction models. We find, in the long run, a 1% increase in the rate of inflation leads to approximately a 0.0046% increase in capacity utilization. The asymmetric error correction structure shows that changes in capacity utilization show significant corrective measures only during booms while changes in inflation correct during both phases of the business cycle with the corrections being stronger during recessions. We also find that, in the short run, changes in the inflation rate do Granger cause capacity utilization while changes in capacity utilization do not Granger cause inflation. The Granger causality from inflation to capacity utilization can be interpreted as supporting recent calls made in the popular press by some economists that it may be desirable for the Federal Reserve Bank to try to induce some inflation in an effort to stimulate the economy. In the second essay, we examine the role of consumer confidence on economic activities like households’ consumption in good and bad economic times. We consider the “news” versus “animal spirit” approach interpretation of consumer confidence. In the wake of the Great Recession of 2008-09, many have called for confidence-boosting policies to help speed up the recovery. A recent study has reinforced these policy calls by showing that the Michigan Consumer Confidence Index contains important information about “news” on future productivity that has long-lasting effects on economic activities like aggregate consumption. Using US data, we show this conclusion is more nuanced when considering an economy that has different potential states. We investigate regime-switching models which use the National Bureau of Economic Research US business cycle expansion and contraction data to create an indicator series that distinguishes bad and good economic times and use this series to investigate impulse responses and variance decompositions. We show the connection between consumer confidence to some types of consumer purchases is important during good economic times but is relatively unimportant during bad economic times. We also use this type of model to investigate the connection between news and consumer confidence and this connection is also shown to be state dependent. In the context of the animal spirits versus news debate, our findings show that during economic expansions, consumer confidence shocks likely reflect news, while during economic contractions, consumer confidence shocks are consistent with animal spirits. These findings also have important implications for recent policy debates which consider whether confidence boosting policies, like raising inflation expectations on big-ticket items such as automobiles or business equipment, would lead to a faster recovery. The third essay investigates expectation shocks and their effect on the economy. For instance, this essay investigates whether the economy responds to expectation shocks in an importantly asymmetric way. A growing literature shows that agents' expectation about the future can lead to boom-bust cycles. These studies so far ignore the transmission effects of expectations on current economic activities across the policy regimes. Using the Survey of Professional Forecasters and Livingstone Survey data, this study empirically investigates the effects of expectation shocks on macroeconomic activities when policy regimes shift. Identifying a structural shock to expectations by using the timing of information in the forecast surveys and actual data releases, we show that the effects of agents' expectations about the future on current macroeconomic activities are asymmetric across the policy regimes. In particular, we find that a perception of good times ahead typically leads to a significant rise in current measures of economic activity in a hawkish regime relative to a dovish regime. We also find that monetary policy's reactions to agents' expectations are asymmetric across the policy regimes. Our findings do not support the views of critics of the central banks, who argued that keeping monetary policy too easy for too long is responsible for fueling the booms. Instead, our findings support the traditional view that a positive (negative) expectation about the future coincides with an anticipatory tightening (easing) of monetary policy.

[pt] Tradicionalmente, muitos trabalhos têm estudado os efeitos das intervenções cambiais esterilizadas nas taxas de câmbio, tanto empiricamente quanto teoricamente, encontrando resultados mistos. Mais recentemente, a literatura de finanças internacionais têm procurado explicar os desvios na Paridade Coberta da Taxa de Juros (PCJ), que vem sendo observado entre as moedas das economias desenvolvidas após a Grande Crise Financeira de 2008. Neste trabalho, ligamos as duas literaturas ao estudar o efeito das intervenções cambiais nos desvios na paridade coberta de juros. Nossa amostra consiste nas intervenções realizadas pelo Banco Central do Brasil entre os anos de 2009 e 2020. Este período contempla o programa de intervenções pré-anunciadas de 2013, implementado no contexto do Taper Tantrum, e que já mostrou ter afetado significantemente as taxas de câmbio (Chamon, Garcia e Souza (2017) ). Para avaliar os efeitos, construímos uma série contrafactual utilizando a metodologia ArCo, desenvolvida por Carvalho, Masini e Medeiros (2018), e também estimando funções impulso resposta utilizando Local Projection, desenvolvida por Jordà (2005). Os resultados indicam que a venda de dólares no mercado futuro aumentam os desvios na PCJ, enquanto que compras de dólares tem o efeito oposto. A oferta de dólares via contratos de recompra diminui os desvios no curto prazo. As intervenções no mercado a vista apresentam resultados inconclusivos. / [en] Traditionally, much has been written about the effects of FX (foreign exchange) sterilized interventions on exchange rates, both theoretically and empirically, with mixed results. More recently, the international finance literature has tried to explain the deviations from the well-known Covered Interest Parity (CIP) condition that have, since the 2008 Great Financial Crisis, arisen among advanced economies currencies. Here, we originally merge these two strands of the literature by analyzing the effects of sterilized FX interventions on the CIP (Covered Interest Parity) deviation. Our sample is composed of Brazilian Central Bank FX interventions between 2009 and 2020. This period contains a major program of announced FX interventions in response to the Taper Tantrum, in 2013, which has already been shown to have significantly affected the level of the exchange rate (Chamon, Garcia, and Souza (2017)). To gauge the effects, we build a counterfactual employing the ArCo methodology, developed by Carvalho, Masini, and Medeiros (2018), and also make use of Jordà (2005) Local Projections. The results indicate that selling US dollars in the futures market increases CIP deviations while buying US dollar futures has the opposite effect. Offering US dollar repo credit lines points to a short-lived decrease in the deviation. The number of sterilized sales or purchases of spot currency seems not to be high enough to lead to conclusive results.

Atualmente diversos sistemas sensoriais fornecem dados em fluxos e essas observações medidas são frequentemente de alta dimensionalidade, ou seja, o número de variáveis medidas é grande, e as observações chegam em sequência. Este é, em particular, o caso de sistemas de visão em robôs. Aprendizagem supervisionada e não-supervisionada com esses fluxos de dados é um desafio, porque o algoritmo deve ser capaz de aprender com cada observação e depois descartá-la antes de considerar a próxima, mas diversos métodos requerem todo o conjunto de dados a fim de estimar seus parâmetros e, portanto, não são adequados para aprendizagem em tempo real. Além disso, muitas abordagens sofrem com a denominada maldição da dimensionalidade (BELLMAN, 1961) e não conseguem lidar com dados de entrada de alta dimensionalidade. Para superar os problemas descritos anteriormente, este trabalho propõe um novo modelo de rede neural probabilístico e incremental, denominado Local Projection Incremental Gaussian Mixture Network (LP-IGMN), que é capaz de realizar aprendizagem perpétua com dados de alta dimensionalidade, ou seja, ele pode aprender continuamente considerando a estabilidade dos parâmetros do modelo atual e automaticamente ajustar sua topologia levando em conta a fronteira do subespaço encontrado por cada neurônio oculto. O método proposto pode encontrar o subespaço intrísico onde os dados se localizam, o qual é denominado de subespaço principal. Ortogonal ao subespaço principal, existem as dimensões que são ruidosas ou que carregam pouca informação, ou seja, com pouca variância, e elas são descritas por um único parâmetro estimado. Portanto, LP-IGMN é robusta a diferentes fontes de dados e pode lidar com grande número de variáveis ruidosas e/ou irrelevantes nos dados medidos. Para avaliar a LP-IGMN nós realizamos diversos experimentos usando conjunto de dados simulados e reais. Demonstramos ainda diversas aplicações do nosso método em tarefas de reconhecimento de imagens. Os resultados mostraram que o desempenho da LP-IGMN é competitivo, e geralmente superior, com outras abordagens do estado da arte, e que ela pode ser utilizada com sucesso em aplicações que requerem aprendizagem perpétua em espaços de alta dimensionalidade. / Nowadays several sensory systems provide data in ows and these measured observations are frequently high-dimensional, i.e., the number of measured variables is large, and the observations are arriving in a sequence. This is in particular the case of robot vision systems. Unsupervised and supervised learning with such data streams is challenging, because the algorithm should be capable of learning from each observation and then discard it before considering the next one, but several methods require the whole dataset in order to estimate their parameters and, therefore, are not suitable for online learning. Furthermore, many approaches su er with the so called curse of dimensionality (BELLMAN, 1961) and can not handle high-dimensional input data. To overcome the problems described above, this work proposes a new probabilistic and incremental neural network model, called Local Projection Incremental Gaussian Mixture Network (LP-IGMN), which is capable to perform life-long learning with high-dimensional data, i.e., it can continuously learn considering the stability of the current model's parameters and automatically adjust its topology taking into account the subspace's boundary found by each hidden neuron. The proposed method can nd the intrinsic subspace where the data lie, which is called the principal subspace. Orthogonal to the principal subspace, there are the dimensions that are noisy or carry little information, i.e., with small variance, and they are described by a single estimated parameter. Therefore, LP-IGMN is robust to di erent sources of data and can deal with large number of noise and/or irrelevant variables in the measured data. To evaluate LP-IGMN we conducted several experiments using simulated and real datasets. We also demonstrated several applications of our method in image recognition tasks. The results have shown that the LP-IGMN performance is competitive, and usually superior, with other stateof- the-art approaches, and it can be successfully used in applications that require life-long learning in high-dimensional spaces.

Atualmente diversos sistemas sensoriais fornecem dados em fluxos e essas observações medidas são frequentemente de alta dimensionalidade, ou seja, o número de variáveis medidas é grande, e as observações chegam em sequência. Este é, em particular, o caso de sistemas de visão em robôs. Aprendizagem supervisionada e não-supervisionada com esses fluxos de dados é um desafio, porque o algoritmo deve ser capaz de aprender com cada observação e depois descartá-la antes de considerar a próxima, mas diversos métodos requerem todo o conjunto de dados a fim de estimar seus parâmetros e, portanto, não são adequados para aprendizagem em tempo real. Além disso, muitas abordagens sofrem com a denominada maldição da dimensionalidade (BELLMAN, 1961) e não conseguem lidar com dados de entrada de alta dimensionalidade. Para superar os problemas descritos anteriormente, este trabalho propõe um novo modelo de rede neural probabilístico e incremental, denominado Local Projection Incremental Gaussian Mixture Network (LP-IGMN), que é capaz de realizar aprendizagem perpétua com dados de alta dimensionalidade, ou seja, ele pode aprender continuamente considerando a estabilidade dos parâmetros do modelo atual e automaticamente ajustar sua topologia levando em conta a fronteira do subespaço encontrado por cada neurônio oculto. O método proposto pode encontrar o subespaço intrísico onde os dados se localizam, o qual é denominado de subespaço principal. Ortogonal ao subespaço principal, existem as dimensões que são ruidosas ou que carregam pouca informação, ou seja, com pouca variância, e elas são descritas por um único parâmetro estimado. Portanto, LP-IGMN é robusta a diferentes fontes de dados e pode lidar com grande número de variáveis ruidosas e/ou irrelevantes nos dados medidos. Para avaliar a LP-IGMN nós realizamos diversos experimentos usando conjunto de dados simulados e reais. Demonstramos ainda diversas aplicações do nosso método em tarefas de reconhecimento de imagens. Os resultados mostraram que o desempenho da LP-IGMN é competitivo, e geralmente superior, com outras abordagens do estado da arte, e que ela pode ser utilizada com sucesso em aplicações que requerem aprendizagem perpétua em espaços de alta dimensionalidade. / Nowadays several sensory systems provide data in ows and these measured observations are frequently high-dimensional, i.e., the number of measured variables is large, and the observations are arriving in a sequence. This is in particular the case of robot vision systems. Unsupervised and supervised learning with such data streams is challenging, because the algorithm should be capable of learning from each observation and then discard it before considering the next one, but several methods require the whole dataset in order to estimate their parameters and, therefore, are not suitable for online learning. Furthermore, many approaches su er with the so called curse of dimensionality (BELLMAN, 1961) and can not handle high-dimensional input data. To overcome the problems described above, this work proposes a new probabilistic and incremental neural network model, called Local Projection Incremental Gaussian Mixture Network (LP-IGMN), which is capable to perform life-long learning with high-dimensional data, i.e., it can continuously learn considering the stability of the current model's parameters and automatically adjust its topology taking into account the subspace's boundary found by each hidden neuron. The proposed method can nd the intrinsic subspace where the data lie, which is called the principal subspace. Orthogonal to the principal subspace, there are the dimensions that are noisy or carry little information, i.e., with small variance, and they are described by a single estimated parameter. Therefore, LP-IGMN is robust to di erent sources of data and can deal with large number of noise and/or irrelevant variables in the measured data. To evaluate LP-IGMN we conducted several experiments using simulated and real datasets. We also demonstrated several applications of our method in image recognition tasks. The results have shown that the LP-IGMN performance is competitive, and usually superior, with other stateof- the-art approaches, and it can be successfully used in applications that require life-long learning in high-dimensional spaces.

Atualmente diversos sistemas sensoriais fornecem dados em fluxos e essas observações medidas são frequentemente de alta dimensionalidade, ou seja, o número de variáveis medidas é grande, e as observações chegam em sequência. Este é, em particular, o caso de sistemas de visão em robôs. Aprendizagem supervisionada e não-supervisionada com esses fluxos de dados é um desafio, porque o algoritmo deve ser capaz de aprender com cada observação e depois descartá-la antes de considerar a próxima, mas diversos métodos requerem todo o conjunto de dados a fim de estimar seus parâmetros e, portanto, não são adequados para aprendizagem em tempo real. Além disso, muitas abordagens sofrem com a denominada maldição da dimensionalidade (BELLMAN, 1961) e não conseguem lidar com dados de entrada de alta dimensionalidade. Para superar os problemas descritos anteriormente, este trabalho propõe um novo modelo de rede neural probabilístico e incremental, denominado Local Projection Incremental Gaussian Mixture Network (LP-IGMN), que é capaz de realizar aprendizagem perpétua com dados de alta dimensionalidade, ou seja, ele pode aprender continuamente considerando a estabilidade dos parâmetros do modelo atual e automaticamente ajustar sua topologia levando em conta a fronteira do subespaço encontrado por cada neurônio oculto. O método proposto pode encontrar o subespaço intrísico onde os dados se localizam, o qual é denominado de subespaço principal. Ortogonal ao subespaço principal, existem as dimensões que são ruidosas ou que carregam pouca informação, ou seja, com pouca variância, e elas são descritas por um único parâmetro estimado. Portanto, LP-IGMN é robusta a diferentes fontes de dados e pode lidar com grande número de variáveis ruidosas e/ou irrelevantes nos dados medidos. Para avaliar a LP-IGMN nós realizamos diversos experimentos usando conjunto de dados simulados e reais. Demonstramos ainda diversas aplicações do nosso método em tarefas de reconhecimento de imagens. Os resultados mostraram que o desempenho da LP-IGMN é competitivo, e geralmente superior, com outras abordagens do estado da arte, e que ela pode ser utilizada com sucesso em aplicações que requerem aprendizagem perpétua em espaços de alta dimensionalidade. / Nowadays several sensory systems provide data in ows and these measured observations are frequently high-dimensional, i.e., the number of measured variables is large, and the observations are arriving in a sequence. This is in particular the case of robot vision systems. Unsupervised and supervised learning with such data streams is challenging, because the algorithm should be capable of learning from each observation and then discard it before considering the next one, but several methods require the whole dataset in order to estimate their parameters and, therefore, are not suitable for online learning. Furthermore, many approaches su er with the so called curse of dimensionality (BELLMAN, 1961) and can not handle high-dimensional input data. To overcome the problems described above, this work proposes a new probabilistic and incremental neural network model, called Local Projection Incremental Gaussian Mixture Network (LP-IGMN), which is capable to perform life-long learning with high-dimensional data, i.e., it can continuously learn considering the stability of the current model's parameters and automatically adjust its topology taking into account the subspace's boundary found by each hidden neuron. The proposed method can nd the intrinsic subspace where the data lie, which is called the principal subspace. Orthogonal to the principal subspace, there are the dimensions that are noisy or carry little information, i.e., with small variance, and they are described by a single estimated parameter. Therefore, LP-IGMN is robust to di erent sources of data and can deal with large number of noise and/or irrelevant variables in the measured data. To evaluate LP-IGMN we conducted several experiments using simulated and real datasets. We also demonstrated several applications of our method in image recognition tasks. The results have shown that the LP-IGMN performance is competitive, and usually superior, with other stateof- the-art approaches, and it can be successfully used in applications that require life-long learning in high-dimensional spaces.

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