Control del proceso de nitrificación

The research methodology in this study is divided into four stages. The stages are driven by the normal distribution theorem coupled with the aforementioned research philosophy, approach, design and strategy. In this thesis, the application of normal distribution theory is allowed through normality tests, descriptive statistics with bootstrap intervals, control limits, bias-corrected and accelerated bootstrapping. The case studies used in this research include the last two housing bubbles that occurred in the UK housing market. These occurred in the periods of 1986-1989 (Scott, 1996; Fraser, 1993; Kuenzel & Bjornbak, 2008; Dolphin & Griffith, 2011) and 2001/2-2007 (Rapp, 2009; Clark et al., 2010; Kuenzel & Bjornbak, 2008; Dolphin & Griffith, 2011).

Using an ex post approach, I seek to confirm whether accepted historic bubble episodes (UK housing bubble cases) are in line with the model’s rule violations. The data used in this study have been collected from various national sources such as Nationwide, Halifax, the Council of Mortgage Lenders, the Department of Communities and Local Governments and the Home Office for National Statistics. The geographical scope of the collected data is limited to the UK at a national level. Further description of the data used is provided in Section 5.3.

5.2.1 Research process

The research process defines the plan for the development of the housing bubble model. It also details and justifies which methods and techniques are used in each stage of the research process in relation to data collection, analysis and validation.

Figure 7. Stages of the research process

Stage 1

• Identification and verification of the variables

• Normality tests, Descriptive statistics-Bootsrapping, Quality contol limits, and Correlation analysis

Stage 2

• Identification of the key constructs of the model

• Case study, Moving averages, AUC-Trapezoid rule and Second law of motion, Correlation, Bootstrapping

Stage 3

• Identification of the model diagnostic rule

• Bias-corrected and accelerated bootstrapping

Stage 4

• Test and implement the model

5.2.1.1 Stage 1: Identification and verification of the variables

The first stage involves a longitudinal time series analysis for the period between 1983 and 2011 using multiple housing variables. This analysis aims to empirically identify and explore whether the selected variables exhibited extreme behaviour during the reported housing bubble periods, so as to consider them as symptoms of the phenomenon in the first place. In other words, the analysis intends to compare whether the growth of such symptoms is associated with the periods of housing bubble formation in the UK and whether the bubble periods differ significantly from non-bubble periods in terms of these variables. The tools used to examine this relationship include normality tests, descriptive statistics with traditional bootstrapping, quality control limits and correlation analysis.

5.2.1.2 Stage 2: Identification of the key constructs of the model

The second part of the research process focuses on establishing the key constructs of our model. These include:

I. Hierarchical order of the variables II. Specific time frame of analysis III. Measurement process

IV. Main Multiplier

For the hierarchical order of the variables, I seek to calculate three components.

The first component is the area under the economic curve (AUC), calculated using the trapezoid rule. The second component is the acceleration rate and involves the difference between the average performance (long-term) and the average abnormal performance (during bubbles). The third component is calculated by applying a bubble factor to the second law of motion. Further details regarding the application and the results of these techniques can be found in Section 7.2.1.

The specific period of analysis is the period that the model considers that a bubble. This period has been established through a comparative analysis relying on UK housing bubble

cases. Further justification for this can be found in Section 7.2.2. The measurement process covers the application of the model and allows for correct data transformation for the key constructs of the model. Further details and explanation regarding these can be found in Sections 7.2.1 and 7.2.3. The main multiplier is the main multiplicand factor in the proposed model. For the selection of the main multiplier variable, I use two methods. The first involves quantitative analysis using existing research tools, and the second includes theoretical reasoning from the literature. The analysis and explanation of that model construct is presented in Section 7.2.4.

5.2.1.3 Stage 3: Identification of the model diagnostic rule

The third part of the research process concentrates on establishing the model rule and proposing the model. The tool used to establish the diagnostic bubble model rule is the bias- corrected and accelerated bootstrapping ( ) method. The rationale for using the method, and further details on this approach can be found in Sections 7.3.3 and 7.3.4.

5.2.1.4 Stage 4: Tests and implementation for the bubble case studies

For the testing procedure of our model, I use the historical data validation (HDV) and event validity (EV) techniques. These techniques aim to test the reliability of our model. HDV involves considering historical input to determine whether the model reproduces real historical output. One example is to determine whether meteorological conditions that have always produced rainy days in reality will produce rainy or sunny days in the model (Kennedy et al. 2005; Sargent, 2005; Yoe, 2012). In this analysis, HDV and EV involve the use of historical bubble input to determine whether the model reproduces an out-of-rule state that in turn corresponds to the bubble component, and whether historical non-bubble input reproduces a state of rule control. Following this method, part of the dataset is used to build the model rule (using the tool) and the remaining datasets are used to determine whether the model represents reality (Kennedy et al., 2005; Sargent, 2005). Figure 8 shows an adapted version of Kennedy et al.’s (2005) verification and validation process for economic model-simulations. It has been modified based on the main question of this thesis. Figure 8 demonstrates the methodology used for its verification and validation. Note that both the

BC_a BC_a

verification of input and the verification of the rule are fundamental parts of the model development process.

Figure 8. Verification and validation process for the bubble model