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Linear Regression

Linear regression belongs to the econometric methods of empirical research, which are applied in almost all sciences. Linear regression is a set of econometric methods of estimating statistical causality between two or more factors (variables of interest). A central assumption of linear regression is the ceteris paribus condition, which means nothing other than “if the conditions are the same” or “if other factors are kept constant“. The literature on linear regression can be found in almost every textbook on statistics, textbooks on econometrics, and research literature on methods of empirical economic research.

Simple Linear Model

In a simple linear model, it is assumed that only one variable $x_{1i}$ has significant impact on variable $y_i$ and all unexlained variance is explained by the error term $e_i$.

y_i=\beta_0 + \beta_1 \cdot x_{1i} + e_i \, \text{or} \, y_i=\beta_0+\sum_{j=1}^{k=1} \beta_i \cdot x_{ji} + e_i

Multi-Regression model

y_i=\beta_0 + \beta_1 \cdot x_{1i} + \beta_2 \cdot x_{2i} + ... + \beta_{k-1} \cdot x_{(k-1)i} + \beta_k \cdot x_{ki} + e_i
y_i=\beta_0+\sum_{j=1}^{k} \beta_i \cdot x_{ji} + e_i

Why is the linear regression method used in science?

Scientists use the regression method to explain the statistical causality between two or more factors so that they can identify the potential statistical correlations in their research question. However, a researcher also wants to test whether statistical estimates reflect reality, or at least whether the observation in the sample of his/her observations reflects reality in the population. Regression methods can be used to calculate or analyze relevant factors in a research question. While regression calculation aims to estimate the relevant coefficients (causality estimators), regression analysis aims to test relevant empirical hypotheses (inferential statistics).

Where is linear regression applied in economics and business administration?

Consider the following example. As an economics student, you are used to reading the following statements in almost all economics and business administration textbooks: “The law of demand says that when prices rise, the demand for a normal good falls”. Where does the statement of the law of demand come from? Can this assertion be proven empirically? When is a good a normal good? Although the answers to these questions can be found in any textbook, the background to their justifications and sometimes incomplete explanations are more likely to be found in empirical research using econometric methods.

Economists work with theoretical models that can be empirically tested to determine the extent to which they reflect your research question in reality. In the case of the law of demand, the Cobb-Douglas model can be applied, which assumes constant elasticity of demand. Here is where the first problem arises. This Cobb-Douglas theoretical model is not linear, as required by linear regression methods, but a non-linear (multiplicative) model. Utilizing the logarithm, however, the Cobb-Douglas model can be transformed into a (log-to-log) linear model (linear transformation). With a sufficient sample, a regression model can be estimated to statistically verify the claims. This example is one of many other applications of empirical analysis to test economic theories.

Simple and multi-regression analysis

In econometric regression analysis, a distinction is made between simple regression analysis and multi-regression analysis. In simple regression analysis, two factors are examined, e.g. a macroeconomic hypothesis could be that domestic consumption (C) has a positive influence on domestic income (Y). The propagated causality is that domestic income depends on domestic consumption – Y(C).

Econometric models start with a simple regression between two variables.

The aim of econometrics is to estimate empirical models that confirm or even refute the propagated causality between domestic consumption and domestic income of a country in the given population. The reverse causality, however, is also possible that domestic consumption tends to depend on domestic income – C(Y). Now, such an analysis, Y(C) and C(Y) takes place under the assumption of the ceteris paribus condition.

The simple regression is then extended by further variables – forming the multi-regression model.

Due to the ceteris paribus condition, the explanatory power of the simple model is limited to explaining the potential causality between domestic consumption and domestic income of a country, but without reference to other potential causalities between other (non-) observable factors, e.g. domestic and foreign investment, exports, imports, government expenditure, savings, taxes, etc. For this reason, the simple regression model is extended. If we now extend the propagated causality to other potential causalities, this results in the multi-regression model, e.g. domestic consumption (C) is influenced by disposable income (income (Y) minus taxes (T)) and other factors, the classical macroeconomic theory of consumption according to Keynes.

C(Y_V)=c_1Y_v+c_0

The Technique of Scientific Writing in Economics

Since I started training individual students from several economic faculties in German Universities about the technique of scientific writing, they asked me where they could find some inspiration to get an appropriate topic for their Bachelor and Master thesis. Depending on the regulations of your university, the supervising professor or lecturer may suggest a specific topic (area of research). Another alternative, the lecturer may give you handouts as an inspiration (starting point) for your research and expect that you will define your own specific research question and then consult with them to fix the final topic. Before you spend a lot of time beating about the bush on how to come up with a convincing draft, your proposal should contain the following ingredients: (research topic, abstract, table of content that shows a clear argumentation line, list of literature sources, model, methodology).

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How to design your Economic Research Topic for your Thesis

To arrive at your final convincing draft, you will need to observe the following sub-processes.

1. Define the Economic Problem in your Thesis

Designing your economic research topic begins with the definition of the economic problem you would like to review and analyze, e. g. such a topic “The impact of global trade imbalances on the European Union (EU)”. In such a topic, the economic problem is the “global trade imbalances”, while the “European Union (EU)” limits the context of the analysis.

2. Construct a Mind-Map for your Economic Problem and Context

At this point you should draw a mind-map depicting two general areas: (a) global trade imbalances and (b) European Union (EU), connected with an arrow from (a) to (b) to resemble (c). This and its impact is your main focus in dealing with the economic problem and the chosen context. Using a mind-map you can develop relevant keywords relating to your topic and potential theories that relate to each topic area, e. g. in (a) global trade imbalances is a macroeconomic topic on international trade, while (b) European Union (EU) is also an economic topic of economic policy (institutions), economic integration, international economics as well as international trade. This should give you a hint about the topics you should be familiar with or the literature you should consult.

3. Create spontaneous Mind-Maps and Literature-based Mind-Maps

Each time you need an idea for your thesis contract a mind-map out of your spontaneous knowledgebase. Afterwards use literature sources to construct literature-based and specific mind-maps of what a certain source delivers to you. Now you should have more than one mind-map that talks about the same topic, but from different perspectives (inspiration). Look for commonalities between your spontaneous ideas and the literature sources. Apply critical thinking: ask yourself, why there are differences and whether other sources may help to bridge the gap or not.

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