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More Than Two Explanatory Variables

in Introductory Econometrics: Intuition, Proof, and Practice

This chapter shows that the addition of a second explanatory variable in Chapter 11 adds only four new things to what there is to know about regression. First, regression uses only the parts of each ... More

This chapter shows that the addition of a second explanatory variable in Chapter 11 adds only four new things to what there is to know about regression. First, regression uses only the parts of each variable that are unrelated to all of the other variables. Second, omitting a variable from the sample relationship that appears in the population relationship almost surely biases our estimates. Third, including an irrelevant variable does not bias estimates but reduces their precision. Fourth, the number of interesting joint tests increases with the number of slopes. All four remain valid when we add additional explanatory variables.Less

Understanding and Interpreting Regression with Two x's

in Introductory Econometrics: Intuition, Proof, and Practice

This chapter begins by deriving the variances of the slopes from a given equation. These are used first to address the issue of multicollinearity. It then turns to the issue of interpreting ... More

This chapter begins by deriving the variances of the slopes from a given equation. These are used first to address the issue of multicollinearity. It then turns to the issue of interpreting regression results. The slopes obtained when the sum of squared errors for the regression of the same equation is minimized are best linear unbiased (BLU) estimates of the population coefficients. If the population relationship includes two explanatory variables, the precision of these slopes depends heavily on the extent to which the two explanatory variables are related. Including an irrelevant variable is inefficient, but does not create bias. Everything that was done in Chapters 8 through 10 holds with two explanatory variables, either exactly or with minor, sensible extensions.Less

What If There Is More Than One X?

in Introductory Econometrics: Intuition, Proof, and Practice

This chapter shows that if the population relationship includes two explanatory variables, but the sample regression contains only one, then the estimate of the effect of the included variable is ... More

This chapter shows that if the population relationship includes two explanatory variables, but the sample regression contains only one, then the estimate of the effect of the included variable is almost surely biased. The best remedy is to include the omitted variable in the sample regression. Minimizing the sum of squared errors from a regression with two explanatory variables yields two slopes, each of which represents the relationship between the parts of the dependent variable and the associated explanatory variable that are not related to the other explanatory variable. These slopes are unbiased estimators of the population coefficients.Less

Regression models

Bendix Carstensen

in Epidemiology with R

This chapter evaluates regression models, focusing on the normal linear regression model. The normal linear regression model establishes a relationship between a quantitative response (also called ... More

This chapter evaluates regression models, focusing on the normal linear regression model. The normal linear regression model establishes a relationship between a quantitative response (also called outcome or dependent) variable, assumed to be normally distributed, and one or more explanatory (also called regression, predictor, or independent) variables about which no distributional assumptions are made. The model is usually referred to as 'the general linear model'. The chapter then differentiates between simple linear regression and multiple regression. The term 'simple linear regression' covers the regression model where there is one response variable and one explanatory variable, assuming a linear relationship between the two. The chapter also discusses the model formulae in R; generalized linear models; collinearity and aliasing; and logarithmic transformations.Less

Categorical Dependent Variables

in Introductory Econometrics: Intuition, Proof, and Practice

The previous chapters focused on how one or more explanatory variables contributed to the value of a dependent variable. It was assumed that the dependent variable was continuous and reported with ... More

The previous chapters focused on how one or more explanatory variables contributed to the value of a dependent variable. It was assumed that the dependent variable was continuous and reported with precision. However, this is not the case for all dependent variables. Suppose the outcome in which we are interested is not continuous or, at least, is not measured continuously. How can we identify the extent to which such outcomes depend on their determinants? This chapter introduces methods that can be used for this purpose. The dependent variables of interest here are categorical rather than continuous. Their outcomes are categories, such as employed or not employed, a college graduate or not a college graduate, a union member or not a union member, rather than quantities. These dependent variables are also discrete because their possible outcomes are distinct from each other in character rather than varying in quantity alone.Less

Explaining Why Firms May Become Alphas or Betas

Raj Chari

in Life After Privatization

Offering models to guide the book’s empirical investigation (in Chapters 3, 4, and 5), this chapter outlines the explanatory variables (factors) that may explain why a firm becomes an Alpha or a ... More

Offering models to guide the book’s empirical investigation (in Chapters 3, 4, and 5), this chapter outlines the explanatory variables (factors) that may explain why a firm becomes an Alpha or a Beta. The first includes the firm’s economic conditions when initially sold and the role of the state during privatization. The second includes factors of influence after privatization, highlighting two broad categories. First are those internal to the firm: the firm’s competitiveness, the managers’ goals, and the role of the shareholders. Second are those more political factors external to the firm: the impact of liberalization; the roles of states in crafting M&A; regulatory decisions on deals; and firm lobbying. The chapter closes by outlining the methods of analysis used in the empirical chapters to determine whether or not a firm is an Alpha or a Beta and to gauge which of the factors are of salience.Less

Regression

Andy Hector

in The New Statistics with R: An Introduction for Biologists

This chapter extends the use of linear models to relationships with continuous explanatory variables, in other words, linear regression. The goal of the worked example (on timber hardness data) given ... More

This chapter extends the use of linear models to relationships with continuous explanatory variables, in other words, linear regression. The goal of the worked example (on timber hardness data) given in detail in this chapter is prediction, not hypothesis testing. Confidence intervals and prediction intervals are explained. Graphical approaches to checking the assumptions of linear-model analysis are explored in further detail. The effects of transformations on linearity, normality, and equality of variance are investigated.Less

Linear Regression

Andy Hector

in The New Statistics with R: An Introduction for Biologists

This chapter extends the use of linear models to relationships with continuous explanatory variables, in other words, linear regression. The goal of the worked example (on wood hardness data) given ... More

This chapter extends the use of linear models to relationships with continuous explanatory variables, in other words, linear regression. The goal of the worked example (on wood hardness data) given in detail in this chapter is prediction, not hypothesis testing. Confidence intervals and prediction intervals are explained. Graphical approaches to checking the assumptions of linear model analysis are explored in further detail. The effects of transformations on linearity, normality, and equality of variance are investigated.Less

Understanding Policy Reversals and Policy Stability

Wolfgang C. Müller and Paul W. Thurner (eds)

in The Politics of Nuclear Energy in Western Europe

This chapter discusses structural, institutional, and situational factors that exercise influence on nuclear energy policy decisions. It reviews the respective literatures and introduces the ... More

This chapter discusses structural, institutional, and situational factors that exercise influence on nuclear energy policy decisions. It reviews the respective literatures and introduces the dependent variable, i.e. nuclear energy policy reversals. Building in particular on the work of Kitschelt (1986) and Midttun and Rucht (1994), the chapter then discusses the explanatory variables that potentially drive such changes: anti-nuclear movements, public opinion, the systems’ electoral and federal openness, political parties’ vote-seeking, principled ideological goals, or office-seeking, new policy challenges in terms of energy policy and climate change concerns, nuclear accidents, and path dependence due to the countries investment in nuclear energy. Hypotheses are formulated for how these factors impact nuclear energy policy-making.Less

Linear Models

Andy Hector

in The New Statistics with R: An Introduction for Biologists

The last chapter conducted a simple analysis of Darwin’s maize data using R as an oversized pocket calculator to work out confidence intervals ‘by hand’. This is a simple way to learn about analysis ... More

The last chapter conducted a simple analysis of Darwin’s maize data using R as an oversized pocket calculator to work out confidence intervals ‘by hand’. This is a simple way to learn about analysis and good for demystifying the process, but it is inefficient. Instead, we want to take advantage of the more sophisticated functions that R provides that are designed to perform linear-model analysis. This chapter explores those functions by repeating and extending the analysis of Darwin’s maize data.Less