Browsing by Author "Park, Joon Y."
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Item Econometric analysis of ARCH models with persistent covariates(2006) Han, Heejoon; Park, Joon Y.We consider a volatility model, named ARCH-NNH model, that is specifically an ARCH process with a nonlinear function of a persistent, integrated or nearly integrated, explanatory variable. We first establish the asymptotic theories showing that the time series properties of our model successfully describe stylized facts about volatility in financial time series. Due to persistent covariates, the model generates time series showing the long memory property in volatility and leptokurtosis which are commonly observed in speculative return series. Next, we derive the asymptotic distribution theory of the maximum likelihood estimator in our model. In particular, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator in the model, which ensure the standard inference procedure is valid in our model. Additionally, we conduct misspecification analysis and provide an explanation of the commonly observed IGARCH behavior of financial time series. Our theory shows that the IGARCH behavior could be spurious and could be the result of ignoring persistent covariates in ARCH type models. Finally, we present two empirical applications and a forecast evaluation of our model. Both empirical applications show that our model fits the data very well, and the estimation results confirm the findings of other literature. It is shown that the default premium (the yield spread between Baa and Aaa corporate bonds) predicts stock return volatility, and the interest rate differential between two countries accounts for exchange rate return volatility. The forecast evaluation shows that our model generally performs better than other standard volatility models at relatively lower frequencies.Item Essays on the modeling of time-varying densities(2007) Qian, Junhui; Park, Joon Y.We present three essays on the econometric modeling of time-varying densities. In all three studies, we treat density functions themselves as random elements taking values in the Hilbert space of square integrable functions. The first essay introduces functional autoregression of one sequence of time-varying density functions. The second essay develops functional regression of one sequence of densities, the regressand, on another sequence of densities, the regressor. The third essay is concerned with the regression of a sequence of scalar random variables on a sequence of densities. For all three models, we present methods of estimation, which all involve the solution of an ill-posed inverse problem, and show asymptotic consistency of our estimators. We also outline hypotheses testing strategies based on each model and show asymptotic distribution of test statistics that we have developed. Possible applications of each model in economics and finance are indicated in each essay. In particular, we apply the functional autoregression model to the analysis of intraday return distributions of S&P 500 index and US/UK exchange rate and find that the functional method offers an outstanding performance in out-of-sample forecasting and a unique way to test on the moment dependence structure of time-varying distributions.Item Essays on time series: Time change and applications to testing, estimation and inference in continuous time models(2007) Vasudev, Rahul; Park, Joon Y.In Chapter 1, I develop a test for the martingale hypothesis using the fact that a continuous martingale is time-deformed browninan motion, where the deforming process is quadratic variation of the martingale. Sampling a martingale at equal increases in quadratic variation and taking first differences, we may obtain variables that are identically, independently distributed as normal. I propose tests that involve thus transforming a sample and testing for the martinagale hypothesis by using the distance between the empirical distribution of the transformed variables and the standard normal distribution using Kolmogorov-Smirnov and Cramer-von Mises statistics. Asymptotics in this setting involve sampling the process more frequently over a given sampling horizon in addition to sampling over longer horizons. Simulations show that the size and power performance of the tests is rather satisfactory. The test is employed on a variety of financial futures to test for a risk premium. In Chapter 2, I develop a method of estimating drift terms in diffusion models that involves transforming the discretely observed sample and performing a least squares regression. This method allows for the estimation of a parametrically specified drift term when the diffusion term is unspecified. The procedure involves sampling data at equal increases in estimated quadratic variation and taking first differences. The transformed variables can be written as the sum of two quantities, a function of the drift parameters and, a regression error that forms a martingale difference sequence and is homoscedastic in the limit when the sampling interval is small. Thus the procedure relies on asymptotics of a small sampling interval as well as a large sample. Simulations show that the procedure outperforms ordinary least squares for nonstationary processes. In Chapter 3, I revisit the testing problem studied in Chapter 1 and consider a test based on second moments instead of distrbutional distance. Simulations show that for mean-reverting processes, the test is more powerful than the tests of Chapter 1. Further, I present a time-change result for discrete martingales, and show that discrete martingales sampled at equal increases in quadratic variation and differenced are homoscedastic to an error.Item Three essays on time series with nonstandard nonstationary models(2005) Miller, James Isaac; Park, Joon Y.In the first chapter; we consider nonlinear transformations of random walks driven by thick-tailed innovations with infinite means or variances. In particular, we show how nonlinearity, nonstationarity, and thick tails interact to generate persistency in memory, and we clearly demonstrate that this triad may generate a broad spectrum of persistency patterns. Time series generated by nonlinear transformations of random walks with thick-tailed innovations have asymptotic autocorrelations that decay very slowly as the number of lags increases or do not even decay at all and remain constant at all lags. We also discuss nonlinear regression asymptotics when the regressor is observable and an alternative regression technique when it is unobservable. We use our model to analyze two empirical applications: exchange rates governed by a target zone and electricity price spikes driven by capacity shortfalls. In the second chapter of the dissertation, we show that typical tests for purchasing power parity using target zone exchange rates are inherently misspecified. The real exchange rate cannot be mean-reverting by construction, since the nominal exchange rate is generated by a nonlinear transformation of a nonstationary economic fundamental. As an alternative, we propose basing the real exchange rate (and thus the test) on conditional expectations of the fundamental itself. Real exchange rates based on the true fundamental may in fact exhibit mean reversion, if the long-run purchasing power parity hypothesis holds. The final chapter investigates the statistical properties of the Kalman filter for state space models including integrated time series. In particular, we derive the full asymptotics of maximum likelihood estimation for a prototypical class of such models: those with a single latent common stochastic trend. We demonstrate the utility of the state space model by extracting a common stochastic trend in three empirical analyses: interest rates, stock return volatility and trading volume, and a stock price index.