In this article, we propose an overview of missing data problem, introduce three missing data mechanisms and study general solutions to them when estimating a linear regression equation. When we have partly missing data, there are two common ways to solve this problem. One way is to ignore those records with missing values. Another method is to impute those observations being missed. Imputation methods arepreferred since they provide full datasets. We observed that there is not a general imputation solution in missing not at random (MNAR) mechanism. In order to check the performance of existing imputation methods in a regression model, a simulation study is set up. Listwise deletion, simple imputation and multiple imputation are selected into comparison which focuses on the effect on parameter estimates and standard errors. The simulation results illustrate that the listwise deletion provides reliable parameter estimates. Simple imputation performs better than multiple imputation in a model with a high determination coefficient. Multiple imputation,which offers a suitable solution for missing at random (MAR), is not valid for MNAR.

By means of a Monte Carlo study, this paper investigates the relative performance of Listwise Deletion, the EM-algorithm and the default algorithm in the MICE-package for R (PMM) in estimating regression coefficients under a left truncated covariate with missing values. The intention is to investigate whether the three frequently used missing data techniques are robust against left truncation when missing values are MCAR or MAR. The results suggest that no technique is superior overall in all combinations of factors studied. The EM-algorithm is unaffected by left truncation under MCAR but negatively affected by strong left truncation under MAR. Compared to the default MICE-algorithm, the performance of EM is more stable across distributions and combinations of sample size and missing rate. The default MICE-algorithm is improved by left truncation but is sensitive to missingness pattern and missing rate. Compared to Listwise Deletion, the EM-algorithm is less robust against left truncation when missing values are MAR. However, the decline in performance of the EM-algorithm is not large enough for the algorithm to be completely outperformed by Listwise Deletion, especially not when the missing rate is moderate. Listwise Deletion might be robust against left truncation but is inefficient.

This thesis investigates the impact of fragmentation in the ownership of complementary patents or patent thickets on firms' market value. This question is motivated by the increase in the patent ownership fragmentation following the pro-patent shifts in the US since 1982. The first chapter uses panel data on patenting US manufacturing firms from 1979 to 1996, and estimates the impact of patent thickets on firms' market value. I find that patent thickets lower firms' market value, and firms with a large patent portfolio size experience a smaller negative effect from their thickets. Moreover, no systematic difference exists in the impact of patent thickets on firms' market value over time. The second chapter extends this analysis to account for the indirect impacts of patent thickets on firms' market value. These indirect effects arise through the effects of patent thickets on firms' R\&D and patenting activities. Using panel data on US manufacturing firms from 1979 to 1996, I estimate the impact of patent thickets on market value, R\&D, and patenting as well as the impacts of R\&D and patenting on market value. Employing these estimates, I determine the direct, indirect, and total impacts of patent thickets on market value. I find that patent thickets decrease firms' market value, while I hold the firms’ R\&D and patenting activities constant. I find no evidence of a change in R\&D due to patent thickets. However, there is evidence of defensive patenting (an increase in patenting attributed to thickets), which helps to reduce the direct negative impact of patent thickets on market value. The data sets used in Chapters 1 and 2 have a number of missing observations on regressors. The commonly used methods to manage missing observations are the listwise deletion (complete case) and the indicator methods. Studies on the statistical properties of these methods suggest a smaller bias using the listwise deletion method. Employing Monte Carlo simulations, Chapter 3 examines the properties of these methods, and finds that in some cases the listwise deletion estimates have larger biases than indicator estimates. This finding suggests that interpreting estimates arrived at with either approach requires caution.

This thesis investigates the impact of fragmentation in the ownership of complementary patents or patent thickets on firms' market value. This question is motivated by the increase in the patent ownership fragmentation following the pro-patent shifts in the US since 1982. The first chapter uses panel data on patenting US manufacturing firms from 1979 to 1996, and estimates the impact of patent thickets on firms' market value. I find that patent thickets lower firms' market value, and firms with a large patent portfolio size experience a smaller negative effect from their thickets. Moreover, no systematic difference exists in the impact of patent thickets on firms' market value over time. The second chapter extends this analysis to account for the indirect impacts of patent thickets on firms' market value. These indirect effects arise through the effects of patent thickets on firms' R\&D and patenting activities. Using panel data on US manufacturing firms from 1979 to 1996, I estimate the impact of patent thickets on market value, R\&D, and patenting as well as the impacts of R\&D and patenting on market value. Employing these estimates, I determine the direct, indirect, and total impacts of patent thickets on market value. I find that patent thickets decrease firms' market value, while I hold the firms’ R\&D and patenting activities constant. I find no evidence of a change in R\&D due to patent thickets. However, there is evidence of defensive patenting (an increase in patenting attributed to thickets), which helps to reduce the direct negative impact of patent thickets on market value. The data sets used in Chapters 1 and 2 have a number of missing observations on regressors. The commonly used methods to manage missing observations are the listwise deletion (complete case) and the indicator methods. Studies on the statistical properties of these methods suggest a smaller bias using the listwise deletion method. Employing Monte Carlo simulations, Chapter 3 examines the properties of these methods, and finds that in some cases the listwise deletion estimates have larger biases than indicator estimates. This finding suggests that interpreting estimates arrived at with either approach requires caution.

This thesis provides an introduction to methods for handling missing data. A thorough review of earlier methods and the development of the field of missing data is provided. The thesis present the methods suggested in today’s literature, multiple imputation and maximum likelihood estimation. A simulation study is performed to see if there are circumstances in small samples when any of the two methods are to be preferred. To show the importance of handling missing data, multiple imputation and maximum likelihood are compared to listwise deletion. The results from the simulation study does not show any crucial differences between multiple imputation and maximum likelihood when it comes to point estimates. Some differences are seen in the estimation of the confidence intervals, talking in favour of multiple imputation. The difference is decreasing with an increasing sample size and more studies are needed to draw definite conclusions. Further, the results shows that listwise deletion lead to biased estimations under a missing at random mechanism. The methods are also applied to a real dataset, the Swedish enrollment registry, to show how the methods work in a practical application.

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