Diagnostic Tests Based on Quantile Residuals for Nonlinear Time Series Models

This thesis studies quantile residuals and uses different methodologies to develop test statistics that are applicable in evaluating linear and nonlinear time series models based on continuous distributions. Models based on mixtures of distributions are of special interest because it turns out that for those models traditional residuals, often referred to as Pearson's residuals, are not appropriate. As such models have become more and more popular in practice, especially with financial time series data there is a need for reliable diagnostic tools that can be used to evaluate them. The aim of the thesis is to show how such diagnostic tools can be obtained and used in model evaluation. The quantile residuals considered here are defined in such a way that, when the model is correctly specified and its parameters are consistently estimated, they are approximately independent with standard normal distribution. All the tests derived in the thesis are pure significance type tests and are theoretically sound in that they properly take the uncertainty caused by parameter estimation into account. -- In Chapter 2 a general framework based on the likelihood function and smooth functions of univariate quantile residuals is derived that can be used to obtain misspecification tests for various purposes. Three easy-to-use tests aimed at detecting non-normality, autocorrelation, and conditional heteroscedasticity in quantile residuals are formulated. It also turns out that these tests can be interpreted as Lagrange Multiplier or score tests so that they are asymptotically optimal against local alternatives. Chapter 3 extends the concept of quantile residuals to multivariate models. The framework of Chapter 2 is generalized and tests aimed at detecting non-normality, serial correlation, and conditional heteroscedasticity in multivariate quantile residuals are derived based on it. Score test interpretations are obtained for the serial correlation and conditional heteroscedasticity tests and in a rather restricted special case for the normality test. In Chapter 4 the tests are constructed using the empirical distribution function of quantile residuals. So-called Khmaladze s martingale transformation is applied in order to eliminate the uncertainty caused by parameter estimation. Various test statistics are considered so that critical bounds for histogram type plots as well as Quantile-Quantile and Probability-Probability type plots of quantile residuals are obtained. Chapters 2, 3, and 4 contain simulations and empirical examples which illustrate the finite sample size and power properties of the derived tests and also how the tests and related graphical tools based on residuals are applied in practice.

Studies on Binary Time Series Models with Applications to Empirical Macroeconomics and Finance

This thesis studies binary time series models and their applications in empirical macroeconomics and finance. In addition to previously suggested models, new dynamic extensions are proposed to the static probit model commonly used in the previous literature. In particular, we are interested in probit models with an autoregressive model structure. In Chapter 2, the main objective is to compare the predictive performance of the static and dynamic probit models in forecasting the U.S. and German business cycle recession periods. Financial variables, such as interest rates and stock market returns, are used as predictive variables. The empirical results suggest that the recession periods are predictable and dynamic probit models, especially models with the autoregressive structure, outperform the static model. Chapter 3 proposes a Lagrange Multiplier (LM) test for the usefulness of the autoregressive structure of the probit model. The finite sample properties of the LM test are considered with simulation experiments. Results indicate that the two alternative LM test statistics have reasonable size and power in large samples. In small samples, a parametric bootstrap method is suggested to obtain approximately correct size. In Chapter 4, the predictive power of dynamic probit models in predicting the direction of stock market returns are examined. The novel idea is to use recession forecast (see Chapter 2) as a predictor of the stock return sign. The evidence suggests that the signs of the U.S. excess stock returns over the risk-free return are predictable both in and out of sample. The new "error correction" probit model yields the best forecasts and it also outperforms other predictive models, such as ARMAX models, in terms of statistical and economic goodness-of-fit measures. Chapter 5 generalizes the analysis of univariate models considered in Chapters 2 4 to the case of a bivariate model. A new bivariate autoregressive probit model is applied to predict the current state of the U.S. business cycle and growth rate cycle periods. Evidence of predictability of both cycle indicators is obtained and the bivariate model is found to outperform the univariate models in terms of predictive power.

Bootstrap and Fast Double Bootstrap Tests of Cointegration Rank with Financial Time Series

The likelihood ratio test of cointegration rank is the most widely used test for cointegration. Many studies have shown that its finite sample distribution is not well approximated by the limiting distribution. The article introduces and evaluates by Monte Carlo simulation experiments bootstrap and fast double bootstrap (FDB) algorithms for the likelihood ratio test. It finds that the performance of the bootstrap test is very good. The more sophisticated FDB produces a further improvement in cases where the performance of the asymptotic test is very unsatisfactory and the ordinary bootstrap does not work as well as it might. Furthermore, the Monte Carlo simulations provide a number of guidelines on when the bootstrap and FDB tests can be expected to work well. Finally, the tests are applied to US interest rates and international stock prices series. It is found that the asymptotic test tends to overestimate the cointegration rank, while the bootstrap and FDB tests choose the correct cointegration rank.

The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

Test of the Clustering Hypothesis in the Helsinki Exchanges

This paper investigates the clustering pattern in the Finnish stock market. Using trading volume and time as factors capturing the clustering pattern in the market, the Keim and Madhavan (1996) and the Engle and Russell (1998) model provide the framework for the analysis. The descriptive and the parametric analysis provide evidences that an important determinant of the famous U-shape pattern in the market is the rate of information arrivals as measured by large trading volumes and durations at the market open and close. Precisely, 1) the larger the trading volume, the greater the impact on prices both in the short and the long run, thus prices will differ across quantities. 2) Large trading volume is a non-linear function of price changes in the long run. 3) Arrival times are positively autocorrelated, indicating a clustering pattern and 4) Information arrivals as approximated by durations are negatively related to trading flow.

Bootstrapping the Error Correction Model Cointegration Test

This paper is concerned with using the bootstrap to obtain improved critical values for the error correction model (ECM) cointegration test in dynamic models. In the paper we investigate the effects of dynamic specification on the size and power of the ECM cointegration test with bootstrap critical values. The results from a Monte Carlo study show that the size of the bootstrap ECM cointegration test is close to the nominal significance level. We find that overspecification of the lag length results in a loss of power. Underspecification of the lag length results in size distortion. The performance of the bootstrap ECM cointegration test deteriorates if the correct lag length is not used in the ECM. The bootstrap ECM cointegration test is therefore not robust to model misspecification.

Cognitive deficits and the Paced Auditory Serial Addition Test performance among patients with multiple sclerosis

Multiple sclerosis (MS) is a chronic, inflammatory disease of the central nervous system, characterized especially by myelin and axon damage. Cognitive impairment in MS is common but difficult to detect without a neuropsychological examination. Valid and reliable methods are needed in clinical practice and research to detect deficits, follow their natural evolution, and verify treatment effects. The Paced Auditory Serial Addition Test (PASAT) is a measure of sustained and divided attention, working memory, and information processing speed, and it is widely used in MS patients neuropsychological evaluation. Additionally, the PASAT is the sole cognitive measure in an assessment tool primarly designed for MS clinical trials, the Multiple Sclerosis Functional Composite (MSFC). The aims of the present study were to determine a) the frequency, characteristics, and evolution of cognitive impairment among relapsing-remitting MS patients, and b) the validity and reliability of the PASAT in measuring cognitive performance in MS patients. The subjects were 45 relapsing-remitting MS patients from Seinäjoki Central Hospital, Department of Neurology and 48 healthy controls. Both groups underwent comprehensive neuropsychological assessments, including the PASAT, twice in a one-year follow-up, and additionally a sample of 10 patients and controls were evaluated with the PASAT in serial assessments five times in one month. The frequency of cognitive dysfunction among relapsing-remitting MS patients in the present study was 42%. Impairments were characterized especially by slowed information processing speed and memory deficits. During the one-year follow-up, the cognitive performance was relatively stable among MS patients on a group level. However, the practice effects in cognitive tests were less pronounced among MS patients than healthy controls. At an individual level the spectrum of MS patients cognitive deficits was wide in regards to their characteristics, severity, and evolution. The PASAT was moderately accurate in detecting MS-associated cognitive impairment, and 69% of patients were correctly classified as cognitively impaired or unimpaired when comprehensive neuropsychological assessment was used as a "gold standard". Self-reported nervousness and poor arithmetical skills seemed to explain misclassifications. MS-related fatigue was objectively demonstrated as fading performance towards the end of the test. Despite the observed practice effect, the reliability of the PASAT was excellent, and it was sensitive to the cognitive decline taking place during the follow-up in a subgroup of patients. The PASAT can be recommended for use in the neuropsychological assessment of MS patients. The test is fairly sensitive, but less specific; consequently, the reasons for low scores have to be carefully identified before interpreting them as clinically significant.

The scientific basis and development of a matrix metalloproteinase (MMP) -8 specific chair-side test for monitoring of periodontal health and disease from gingival crevicular fluid

Matrix metalloproteinase (MMP) -8, collagenase-2, is a key mediator of irreversible tissue destruction in chronic periodontitis and detectable in gingival crevicular fluid (GCF). MMP-8 mostly originates from neutrophil leukocytes, the first line of defence cells which exist abundantly in GCF, especially in inflammation. MMP-8 is capable of degrading almost all extra-cellular matrix and basement membrane components and is especially efficient against type I collagen. Thus the expression of MMP-8 in GCF could be valuable in monitoring the activity of periodontitis and possibly offers a diagnostic means to predict progression of periodontitis. In this study the value of MMP-8 detection from GCF in monitoring of periodontal health and disease was evaluated with special reference to its ability to differentiate periodontal health and different disease states of the periodontium and to recognise the progression of periodontitis, i.e. active sites. For chair-side detection of MMP-8 from the GCF or peri-implant sulcus fluid (PISF) samples, a dip-stick test based on immunochromatography involving two monoclonal antibodies was developed. The immunoassay for the detection of MMP-8 from GCF was found to be more suitable for monitoring of periodontitis than detection of GCF elastase concentration or activity. Periodontally healthy subjects and individuals suffering of gingivitis or of periodontitis could be differentiated by means of GCF MMP-8 levels and dipstick testing when the positive threshold value of the MMP-8 chair-side test was set at 1000 µg/l. MMP-8 dipstick test results from periodontally healthy and from subjects with gingivitis were mainly negative while periodontitis patients sites with deep pockets ( 5 mm) and which were bleeding on probing were most often test positive. Periodontitis patients GCF MMP-8 levels decreased with hygiene phase periodontal treatment (scaling and root planing, SRP) and even reduced during the three month maintenance phase. A decrease in GCF MMP-8 levels could be monitored with the MMP-8 test. Agreement between the test stick and the quantitative assay was very good (κ = 0.81) and the test provided a baseline sensitivity of 0.83 and specificity of 0.96. During the 12-month longitudinal maintenance phase, periodontitis patients progressing sites (sites with an increase in attachment loss ≥ 2 mm during the maintenance phase) had elevated GCF MMP-8 levels compared with stable sites. General mean MMP-8 concentrations in smokers (S) sites were lower than in non-smokers (NS) sites but in progressing S and NS sites concentrations were at an equal level. Sites with exceptionally and repeatedly elevated MMP-8 concentrations during the maintenance phase were clustered in smoking patients with poor response to SRP (refractory patients). These sites especially were identified by the MMP-8 test. Subgingival plaque samples from periodontitis patients deep periodontal pockets were examined by polymerase chain reaction (PCR) to find out if periodontal lesions may serve as a niche for Chlamydia pneumoniae. Findings were compared with the clinical periodontal parameters and GCF MMP-8 levels to determine the correlation with periodontal status. Traces of C. pneumoniae were identified from one periodontitis patient s pooled subgingival plaque sample by means of PCR. After periodontal treatment (SRP) the sample was negative for C. pneumoniae. Clinical parameters or biomarkers (MMP-8) of the patient with the positive C. pneumoniae finding did not differ from other study patients. In this study it was concluded that MMP-8 concentrations in GCF of sites from periodontally healthy individuals, subjects with gingivitis or with periodontitis are at different levels. The cut-off value of the developed MMP-8 test is at an optimal level to differentiate between these conditions and can possibly be utilised in identification of individuals at the risk of the transition of gingivitis to periodontitis. In periodontitis patients, repeatedly elevated GCF MMP-8 concentrations may indicate sites at risk of progression of periodontitis as well as patients with poor response to conventional periodontal treatment (SRP). This can be monitored by MMP-8 testing. Despite the lower mean GCF MMP-8 concentrations in smokers, a fraction of smokers sites expressed very high MMP-8 concentrations together with enhanced periodontal activity and could be identified with MMP-8 specific chair-side test. Deep periodontal lesions may be niches for non-periodontopathogenic micro-organisms with systemic effects like C. pneumoniae and possibly play a role in the transmission from one subject to another.

Anthropometric measures of obesity-their association with type 2 diabetes and hypertension across ethnic groups

Clinical trials have shown that weight reduction with lifestyles can delay or prevent diabetes and reduce blood pressure. An appropriate definition of obesity using anthropometric measures is useful in predicting diabetes and hypertension at the population level. However, there is debate on which of the measures of obesity is best or most strongly associated with diabetes and hypertension and on what are the optimal cut-off values for body mass index (BMI) and waist circumference (WC) in this regard. The aims of the study were 1) to compare the strength of the association for undiagnosed or newly diagnosed diabetes (or hypertension) with anthropometric measures of obesity in people of Asian origin, 2) to detect ethnic differences in the association of undiagnosed diabetes with obesity, 3) to identify ethnic- and sex-specific change point values of BMI and WC for changes in the prevalence of diabetes and 4) to evaluate the ethnic-specific WC cutoff values proposed by the International Diabetes Federation (IDF) in 2005 for central obesity. The study population comprised 28 435 men and 35 198 women, ≥ 25 years of age, from 39 cohorts participating in the DECODA and DECODE studies, including 5 Asian Indian (n = 13 537), 3 Mauritian Indian (n = 4505) and Mauritian Creole (n = 1075), 8 Chinese (n =10 801), 1 Filipino (n = 3841), 7 Japanese (n = 7934), 1 Mongolian (n = 1991), and 14 European (n = 20 979) studies. The prevalence of diabetes, hypertension and central obesity was estimated, using descriptive statistics, and the differences were determined with the χ2 test. The odds ratios (ORs) or  coefficients (from the logistic model) and hazard ratios (HRs, from the Cox model to interval censored data) for BMI, WC, waist-to-hip ratio (WHR), and waist-to-stature ratio (WSR) were estimated for diabetes and hypertension. The differences between BMI and WC, WHR or WSR were compared, applying paired homogeneity tests (Wald statistics with 1 df). Hierarchical three-level Bayesian change point analysis, adjusting for age, was applied to identify the most likely cut-off/change point values for BMI and WC in association with previously undiagnosed diabetes. The ORs for diabetes in men (women) with BMI, WC, WHR and WSR were 1.52 (1.59), 1.54 (1.70), 1.53 (1.50) and 1.62 (1.70), respectively and the corresponding ORs for hypertension were 1.68 (1.55), 1.66 (1.51), 1.45 (1.28) and 1.63 (1.50). For diabetes the OR for BMI did not differ from that for WC or WHR, but was lower than that for WSR (p = 0.001) in men while in women the ORs were higher for WC and WSR than for BMI (both p < 0.05). Hypertension was more strongly associated with BMI than with WHR in men (p < 0.001) and most strongly with BMI than with WHR (p < 0.001), WSR (p < 0.01) and WC (p < 0.05) in women. The HRs for incidence of diabetes and hypertension did not differ between BMI and the other three central obesity measures in Mauritian Indians and Mauritian Creoles during follow-ups of 5, 6 and 11 years. The prevalence of diabetes was highest in Asian Indians, lowest in Europeans and intermediate in others, given the same BMI or WC category. The  coefficients for diabetes in BMI (kg/m2) were (men/women): 0.34/0.28, 0.41/0.43, 0.42/0.61, 0.36/0.59 and 0.33/0.49 for Asian Indian, Chinese, Japanese, Mauritian Indian and European (overall homogeneity test: p > 0.05 in men and p < 0.001 in women). Similar results were obtained in WC (cm). Asian Indian women had lower  coefficients than women of other ethnicities. The change points for BMI were 29.5, 25.6, 24.0, 24.0 and 21.5 in men and 29.4, 25.2, 24.9, 25.3 and 22.5 (kg/m2) in women of European, Chinese, Mauritian Indian, Japanese, and Asian Indian descent. The change points for WC were 100, 85, 79 and 82 cm in men and 91, 82, 82 and 76 cm in women of European, Chinese, Mauritian Indian, and Asian Indian. The prevalence of central obesity using the 2005 IDF definition was higher in Japanese men but lower in Japanese women than in their Asian counterparts. The prevalence of central obesity was 52 times higher in Japanese men but 0.8 times lower in Japanese women compared to the National Cholesterol Education Programme definition. The findings suggest that both BMI and WC predicted diabetes and hypertension equally well in all ethnic groups. At the same BMI or WC level, the prevalence of diabetes was highest in Asian Indians, lowest in Europeans and intermediate in others. Ethnic- and sex-specific change points of BMI and WC should be considered in setting diagnostic criteria for obesity to detect undiagnosed or newly diagnosed diabetes.

Geospatial environmental data modelling applications using remote sensing, GIS and spatial statistics

This thesis presents novel modelling applications for environmental geospatial data using remote sensing, GIS and statistical modelling techniques. The studied themes can be classified into four main themes: (i) to develop advanced geospatial databases. Paper (I) demonstrates the creation of a geospatial database for the Glanville fritillary butterfly (Melitaea cinxia) in the Åland Islands, south-western Finland; (ii) to analyse species diversity and distribution using GIS techniques. Paper (II) presents a diversity and geographical distribution analysis for Scopulini moths at a world-wide scale; (iii) to study spatiotemporal forest cover change. Paper (III) presents a study of exotic and indigenous tree cover change detection in Taita Hills Kenya using airborne imagery and GIS analysis techniques; (iv) to explore predictive modelling techniques using geospatial data. In Paper (IV) human population occurrence and abundance in the Taita Hills highlands was predicted using the generalized additive modelling (GAM) technique. Paper (V) presents techniques to enhance fire prediction and burned area estimation at a regional scale in East Caprivi Namibia. Paper (VI) compares eight state-of-the-art predictive modelling methods to improve fire prediction, burned area estimation and fire risk mapping in East Caprivi Namibia. The results in Paper (I) showed that geospatial data can be managed effectively using advanced relational database management systems. Metapopulation data for Melitaea cinxia butterfly was successfully combined with GPS-delimited habitat patch information and climatic data. Using the geospatial database, spatial analyses were successfully conducted at habitat patch level or at more coarse analysis scales. Moreover, this study showed it appears evident that at a large-scale spatially correlated weather conditions are one of the primary causes of spatially correlated changes in Melitaea cinxia population sizes. In Paper (II) spatiotemporal characteristics of Socupulini moths description, diversity and distribution were analysed at a world-wide scale and for the first time GIS techniques were used for Scopulini moth geographical distribution analysis. This study revealed that Scopulini moths have a cosmopolitan distribution. The majority of the species have been described from the low latitudes, sub-Saharan Africa being the hot spot of species diversity. However, the taxonomical effort has been uneven among biogeographical regions. Paper III showed that forest cover change can be analysed in great detail using modern airborne imagery techniques and historical aerial photographs. However, when spatiotemporal forest cover change is studied care has to be taken in co-registration and image interpretation when historical black and white aerial photography is used. In Paper (IV) human population distribution and abundance could be modelled with fairly good results using geospatial predictors and non-Gaussian predictive modelling techniques. Moreover, land cover layer is not necessary needed as a predictor because first and second-order image texture measurements derived from satellite imagery had more power to explain the variation in dwelling unit occurrence and abundance. Paper V showed that generalized linear model (GLM) is a suitable technique for fire occurrence prediction and for burned area estimation. GLM based burned area estimations were found to be more superior than the existing MODIS burned area product (MCD45A1). However, spatial autocorrelation of fires has to be taken into account when using the GLM technique for fire occurrence prediction. Paper VI showed that novel statistical predictive modelling techniques can be used to improve fire prediction, burned area estimation and fire risk mapping at a regional scale. However, some noticeable variation between different predictive modelling techniques for fire occurrence prediction and burned area estimation existed.

Evaluation of Vestibular Function with Visual Feedback Posturography and Motorized Head Impulse Test.

Objectives: To evaluate the applicability of visual feedback posturography (VFP) for quantification of postural control, and to characterize the horizontal angular vestibulo-ocular reflex (AVOR) by use of a novel motorized head impulse test (MHIT). Methods: In VFP, subjects standing on a platform were instructed to move their center of gravity to symmetrically placed peripheral targets as fast and accurately as possible. The active postural control movements were measured in healthy subjects (n = 23), and in patients with vestibular schwannoma (VS) before surgery (n = 49), one month (n = 17), and three months (n = 36) after surgery. In MHIT we recorded head and eye position during motorized head impulses (mean velocity of 170º/s and acceleration of 1 550º/s²) in healthy subjects (n = 22), in patients with VS before surgery (n = 38) and about four months afterwards (n = 27). The gain, asymmetry and latency in MHIT were calculated. Results: The intraclass correlation coefficient for VFP parameters during repeated tests was significant (r = 0.78-0.96; p < 0.01), although two of four VFP parameters improved slightly during five test sessions in controls. At least one VFP parameter was abnormal pre- and postoperatively in almost half the patients, and these abnormal preoperative VFP results correlated significantly with abnormal postoperative results. The mean accuracy in postural control in patients was reduced pre- and postoperatively. A significant side difference with VFP was evident in 10% of patients. In the MHIT, the normal gain was close to unity, the asymmetry in gain was within 10%, and the latency was a mean ± standard deviation 3.4 ± 6.3 milliseconds. Ipsilateral gain or asymmetry in gain was preoperatively abnormal in 71% of patients, whereas it was abnormal in every patient after surgery. Preoperative gain (mean ± 95% confidence interval) was significantly lowered to 0.83 ± 0.08 on the ipsilateral side compared to 0.98 ± 0.06 on the contralateral side. The ipsilateral postoperative mean gain of 0.53 ± 0.05 was significantly different from preoperative gain. Conclusion: The VFP is a repeatable, quantitative method to assess active postural control within individual subjects. The mean postural control in patients with VS was disturbed before and after surgery, although not severely. Side difference in postural control in the VFP was rare. The horizontal AVOR results in healthy subjects and in patients with VS, measured with MHIT, were in agreement with published data achieved using other techniques with head impulse stimuli. The MHIT is a non-invasive method which allows reliable clinical assessment of the horizontal AVOR.

Non-Gaussian Cosmological Perturbations from Hybrid Inflation and Preheating

This thesis consists of four research papers and an introduction providing some background. The structure in the universe is generally considered to originate from quantum fluctuations in the very early universe. The standard lore of cosmology states that the primordial perturbations are almost scale-invariant, adiabatic, and Gaussian. A snapshot of the structure from the time when the universe became transparent can be seen in the cosmic microwave background (CMB). For a long time mainly the power spectrum of the CMB temperature fluctuations has been used to obtain observational constraints, especially on deviations from scale-invariance and pure adiabacity. Non-Gaussian perturbations provide a novel and very promising way to test theoretical predictions. They probe beyond the power spectrum, or two point correlator, since non-Gaussianity involves higher order statistics. The thesis concentrates on the non-Gaussian perturbations arising in several situations involving two scalar fields, namely, hybrid inflation and various forms of preheating. First we go through some basic concepts -- such as the cosmological inflation, reheating and preheating, and the role of scalar fields during inflation -- which are necessary for the understanding of the research papers. We also review the standard linear cosmological perturbation theory. The second order perturbation theory formalism for two scalar fields is developed. We explain what is meant by non-Gaussian perturbations, and discuss some difficulties in parametrisation and observation. In particular, we concentrate on the nonlinearity parameter. The prospects of observing non-Gaussianity are briefly discussed. We apply the formalism and calculate the evolution of the second order curvature perturbation during hybrid inflation. We estimate the amount of non-Gaussianity in the model and find that there is a possibility for an observational effect. The non-Gaussianity arising in preheating is also studied. We find that the level produced by the simplest model of instant preheating is insignificant, whereas standard preheating with parametric resonance as well as tachyonic preheating are prone to easily saturate and even exceed the observational limits. We also mention other approaches to the study of primordial non-Gaussianities, which differ from the perturbation theory method chosen in the thesis work.