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INTRODUCTION It is desirable to analyze count data using a cycle of model specification, estimation, testing, and evaluation. This cycle can go from specific to general models – for example, it can begin with Poisson and then test for the negative binomial – or one can use a general-to-specific approach, beginning with the negative binomial and then testing the restrictions imposed by Poisson. In terms of inclusion of regressors in a given count model, either approach might be taken; for the choice of the count data model itself, other than simple choices such as Poisson or negative binomial, the former approach is most often useful. For example, if the negative binomial model is inadequate, there is a very wide range of models that might be considered, rendering a general-to-specific approach difficult to implement. The preceding two chapters have presented the specification and estimation components of this cycle for cross-section count data. In this chapter we focus on the testing and evaluation aspects of this cycle. This includes residual analysis, goodness-of-fit measures, and model specification tests, in addition to classical statistical inference. Residual analysis, based on a range of definitions of the residual for heteroskedastic data such as counts, is presented in section 5.2. A range of measures of goodness of fit, including pseudo R-squareds and a chi-square goodness-of-fit statistic, is presented in section 5.3. Discrimination among nonnested models is the subject of section 5.4.

OVERVIEW In this chapter we provide an introduction to two common extensions of multiple regression models. The first deals with cross-sectional models where the dependent variable is categorical rather than continuous. The second involves time-series models, where the variables of interest are measured repeatedly over time. Throughout the chapter, we use examples from a variety of research situations to illustrate the important issues that must be addressed in each research situation. EXTENSIONS OF OLS We have come a long way in the understanding and use of regression analysis in political science. We have learned, mathematically, where OLS coefficients come from; we have learned how to interpret those coefficients substantively; and we have learned how to use OLS to control for other possible causes of the dependent variable. In Chapter 10, we introduced dummy variables, having used them as independent variables in our regression models. In this chapter, we extend this focus to research situations in which the dependent variable is a dummy variable. Such situations are common in political science, as many of the dependent variables that we find ourselves interested in – such as, whether or not an individual voted in a particular election, or whether or not two countries engaged in a dispute escalate the situation to open warfare – are dummy variables.

Introduction We have seen in Chapter 4 that the Mean-Variance (M-V) rule can justifiably be employed under investment decision settings, in three distinct cases: 1) when the utility function is quadratic, 2) when distributions of return are normal in the face of risk aversion, and 3) when the variance of rates of return is not too large. The assumption under quadratic preferences is generally not accepted because this function has some well-known drawbacks, which have been discussed in detail in Chapter 4. Employing the M-V rule as an approximation to expected utility, although generally providing an excellent approximation, may raise some objections because the quality of the approximation depends on the data set involved: it may provide an excellent fit in one case and not such a good fit in another case. Therefore, the most compelling theoretical argument for the employment of the M-V rule is the case when the distributions of returns are normal in the face of risk aversion. Although the M-V rule is optimal for all distributions that belong to the elliptic family (discussed later in this chapter), we focus first on the normal distribution because most traditional empirical goodness-of-fit tests are for normality. The crucial question raised in the normal case is whether it is reasonable to assume that price changes or returns are normally distributed. We devote a substantial portion of this chapter to the statistical validity of the normality assumption and the economic consequences to the investor who employs the M-V rule when the distributions of return significantly deviate from normality. We employ various approaches to analyze the normality assumption and the induced economic loss when distributions are not normal, but investors make investment decisions “as if” the distributions are normal. Of course, the closer the empirical distribution to the normal distribution, the smaller the expected loss induced by the employment of the M-V rule.

I previously indicated that extended Poisson and negative binomial models have generally been developed to solve either a distributional or variance problem arising in the base Poisson and NB2 models. We shall discover later that some extended negative binomial models rely on the NB1 parameterization rather than on the NB2, e.g. fixed-effects negative binomial. But, except for a few models, most negative binomial models used in actual research are based on the NB2 model. Changes to the negative binomial variance function were considered in the last chapter. In this chapter, we address the difficulties arising when the zero counts in Poisson and NB2 models differ considerably from model distributional assumptions. In the first section we address models that cannot have zero counts. Zero-truncated count models Many times we are asked to model count data that structurally exclude zero counts. Hospital length-of-stay data are an example of this type of data. When a patient first enters the hospital, the count begins upon registration, with the length of stay given as 1. There can be no 0 days – unless we are describing patients who do not enter the hospital. This latter situation describes a different type of model where there may be two generating processes – one, for example, for patients who may or may not enter the hospital, and another for patients who are admitted. This type of model will be discussed later.

INTRODUCTION The majority of continuous-time rainfall-runoff models can be classified as conceptual, as discussed in Chapter 1 (see also Wheater et al ., 1993; Wheater, 2002). This type of model represents the hydrological processes that are seemingly important in the system using a simplified, conceptual representation (Fig. 8.1). These models have three notable characteristics: (a) their model structure is specified a priori, (b) the hydrological properties of the catchments are represented as parameters, which are generally assumed to be constant during each model application, (c) (at least some of) the model parameters have no direct, physical meaning and are not directly measurable. Therefore model parameters are usually estimated via calibration, using the fit of the model output time series to observed data to provide a measure of goodness of fit. In this chapter we introduce the issues associated with calibration, and recent developments that allow the associated uncertainty to be specified. Finally the application of models to ungauged catchments (regional analysis) is discussed. For a more extensive treatment of these subjects, the reader is referred to Rainfall-Runoff Modelling in Gauged and Ungauged Catchments by Wagener et al . (2004) and also the AGU Monograph on Calibration of Watershed Models (Duan et al ., 2003).

This paper investigates the impact of wage subsidies on skill formulation. We analyze two prototypical models of skill formation: (a) a learning-by-doing model and (b) an on-the-job training model. We develop conditions on the pricing of jobs under which the two models are equivalent. In general they are different and have different implications of wage subsidies on skill formation. On-the-job training models predict that wage subsidies reduce skill formation. Learning-by-doing models predict the opposite. The provisional evidence favors the learning-by-doing model. We apply our estimates to investigate the impact of the EITC on skill formation. We estimate that the EITC reduced the long term wages of participants with low levels of education.Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

The aim in this chapter is to extend the analysis of the relationship between two variables to cover the topic of regression. In this introductory discussion we will deal only with linear (straight-line) relationships between two variables ( bivariate regression ). In chapter 8 this analysis will be extended to include more than two variables ( multiple or multivariate regression ), and non-linear relationships will be discussed in chapter 12. As with the discussion of correlation, problems arising from the use of sample data are deferred until the issues of confidence intervals and hypothesis testing are covered in chapters 5 and 6. The concept of regression In the discussion of correlation in chapter 3 we emphasized that no distinction was made between the two variables, X and Y, and that interchanging them would have no effect on the correlation coefficient. In the present chapter we change procedure and introduce a fundamental distinction between the two variables. Explanatory and dependent variables It will often be the case that we have some theoretical reason to think that movements in one of the variables are influenced by movements in the other. In that case, the convention is to use X for the variable that is having the influence , and Y for the variable that is influenced .

Summary Sex ratio variances can indicate control of sex allocation beyond the production of a certain mean. Furthermore, theory predicts that nonbinomial (nonrandom) offspring-group sex ratio distributions should be selectively favoured in some circumstances. Mechanisms of producing nonbinomial sex ratio variances include the allocation of sex to offspring in fixed or other nonrandom sequences. Variances and sequences are thus linked. In this chapter we review predictions of sex ratio models, consider constraints and expectations arising from particular sex-determination mechanisms and discuss statistical methods for variance and sequence analysis. Introduction In Chapters 3 and 4 variance was treated as a consideration necessary to ensure appropriate data analysis. However, variance of offspring-group sex ratios are of interest in themselves both because they may be under selection and because they can indicate parental control of sex allocation. We begin by reviewing predictions of sex ratio models (section 5.3) and considering constraints and expectations arising from particular sex-determination mechanisms (section 5.4). We then introduce sex ratio ‘precision’ and its measurement (section 5.5) and give methods for statistical analysis of sex ratio variance (section 5.6). Sequences of sex allocation, and their relationships with sex ratio variance, are considered in section 5.7. Variance expectations from optimality theory Optimal sex-allocation theory makes predictions not just about mean sex ratios but also about variances. When mating opportunities extend throughout large populations (panmixis) the sex ratio within a group of offspring is selectively neutral as long as the population sex ratio remains at equilibrium (Kolman 1960).