This paper suggests the use of simple minimum distance methods to estimate restricted cointegrating vectors. The method directly employs minimum distance methods on unrestricted cointegrating matrices estimated in the usual way to estimate restricted parameters which are linearly or nonlinearly related to the unrestricted cointegrating vector coefficients. The limiting distribution of the estimates as well as the usual test for the restrictions are derived. A Monte Carlo experiment is undertaken to examine the effectiveness of these methods for cointegrating vectors.

This paper examines the efficiency of the forward yen/dollar market using micro survey data. Conventional tests of unbiasedness do not correspond directly to the zero-profit condition. Instead, we use the survey data to calculate potential profits of individual forecasters based on a natural trading rule. We find that although the survey data are not the best predictor of future spot rates in terms of typical mean square forecast error criteria, the survey data can be used to obtain on average positive profits. However, these profits are small and highly variable. Similar results are found when we examine profits generated by a trading rule using regression forecasts. The profits are found to be correlated with risk type variables but not other available information

The paper analyzes the impact of the initial observation on the problem of testing for unit roots. To this end, we derive a family of optimal tests that maximize a weighted average power criterion with respect to the initial observation. We then investigate the relationship of this optimal family to unit root tests in an asymptotic framework. We find that many popular unit root tests are closely related to specific members of the optimal family, but the corresponding members employ very different weightings for the initial observation. The popular Dickey-Fuller tests, for instance, are closely related to optimal tests which put a large weight on extreme derivations of the initial observation from the deterministic component, whereas other popular tests put more weight on moderate deviations. At the same time, the power of the various unit root tests varies dramatically with the initial observation. This paper therefore helps to explain the results of the comparative power studies of unit root tests, and allows a much deeper understanding of the merits of particular tests in specific circumstances.

Instability of parametric models is a common problem in many fields of economics. In econometrics, these changes in the underlying data generating process are referred to as structural breaks. Although there is an extensive literature on estimation and statistical tests of structural breaks, existing methods fail to adequately capture a break. This dissertation consists of three papers on developing econometric methods for structural breaks and forecasting.

The first chapter develops a new method in estimating the location of a structural break in a linear model and provide theoretical results and empirical applications of the estimator. In finite sample the conventional least-squares estimates a break occurred at either ends of the sample with high probability, regardless of the true break point. I suggest an estimator of the break point that resolves this pile up issue and thus, provide a more accurate estimate of the break. The second chapter constructs a statistical test to test existence of a structural break when the direction of the parameter shift is known. In practice it is likely that a researcher is interested in testing for a structural break in a particular direction because the direction is known, such as policy change or historical data. We incorporate this information in constructing three tests that have higher power when direction is correctly specified. The last chapter proposes a multi-period forecasting method that is robust to model misspecification. When we are interested in obtaining long horizon ahead forecasts, the direct forecast method is more favorable than the iterated forecast because it is more robust to misspecification. However, direct forecast estimates tend to have jagged shapes across horizons. I use a mechanism analogous to ridge regression on the direct forecast model to maintain robustness while smoothing out erratic estimates.

Existing results on the properties and performance of forecast combinations have been derived in the context of mean squared error loss. Under this loss function empirical studies have generally found that estimates of optimal forecast combination weights lead to higher losses than equally-weighted combined forecasts which in turn outperform the best individual predictions. We show that this and other results can be overturned when asymmetries are introduced in the loss function and the forecast error distribution is skewed. We characterize the optimal combination weights for the most commonly used alternatives to mean squared error loss and demonstrate how the degree of asymmetry in the loss function and skews in the underlying forecast error distribution can significantly change the optimal combination weights. We also propose estimation methods and investigate their small sample properties in simulations and in an inflation forecasting exercise.

$p$-Hacking can undermine the validity of empirical research. The central focus of this dissertation is on analyzing existing and developing new statistical methods for detecting $p$-hacking based on the empirical distribution of reported results across studies.

In Chapter 1 we theoretically analyze the problem of testing for $p$-hacking based on distributions of $p$-values across multiple studies. We provide general results for when such distributions have testable restrictions (are non-increasing) under the null of no $p$-hacking. We find novel additional testable shape restrictions for $p$-values based on $t$-tests. These testable restrictions result in more powerful tests for the null hypothesis of no $p$-hacking. When there is also publication bias, our tests are joint tests for $p$-hacking and publication bias. A reanalysis of two prominent datasets shows the usefulness of our new tests.

Chapter 2 provides a careful understanding of the power of methods used to detect different types of $p$-hacking discussed in Chapter 1. We theoretically study the implications of likely forms of $p$-hacking on the distribution of reported $p$-values and the power of existing methods for detecting it. Power can be quite low, depending crucially on the particular $p$-hacking strategy and the distribution of actual effects tested by the studies. We relate the power of the tests to the costs of $p$-hacking and show that power tends to be larger when $p$-hacking is very costly.

Chapter 3 studies Caliper tests that are widely used to test for the presence of $p$-hacking and publication bias based on the distribution of the $z$-statistics across studies. We show that without additional restrictions on the distribution of true effects, Caliper tests may suffer from substantial size distortions. We propose a modification of the existing Caliper test, referred to as the Robust Caliper test, which is shown to control size irrespective of the true effect distribution. We also propose a way of correcting the regression-based version of the Caliper test that allows for the inclusion of additional covariates.

We derive the family of tests for a unit root with maximal power against a point alternative when an arbitrary number of stationary covariates are modeled with the potentially integrated series. We show that very large power gains are available when such covariates are available. We then derive tests which are simple to construct (involving the running of vector autoregressions) and achieve at a point the power envelopes derived under very general conditions. These tests have excellent properties in small samples. We also show that these are obvious and internally consistent tests to run when identifying structural VAR's using long run restrictions.

We derive the family of tests for a unit root with maximal power against a point alternative when an arbitrary number of stationary covariates are modeled with the potentially integrated series. We show that very large power gains are available when such covariates available. We then derive tests which are simple to construct (involving the running of vector autoregressions) and achieve at a point the power envelopes derived under very general conditions. These tests are excellent properties in small samples. We also show that these are obvious and internally consistent tests to run when identifying structural VAR's using long run restrictions.

This thesis is devoted to designing and analyzing statistical decision rules to improve public policy. The first three chapters study the estimation and inference of welfare-maximizing treatment rules in the context of experimental design, spillover effects, and algorithmic fairness. The last chapter focuses on statistical inference in experiments with multiple policy effects (either of different interventions or on different outcomes).

Specifically, Chapter 1 proposes an experimental design for estimation and inference of welfare-maximizing policies with spillover effects. As a first contribution, we introduce a single-wave experiment to estimate the marginaleffect of a change in treatment probabilities, taking spillover effects into account. Using the marginal effect, we propose a practical test for welfare-maximizing policy. As a second contribution, we design a multiple-wave experiment to estimate treatment rules and maximize welfare. We provide asymptotic and small-sample guarantees of the procedure and study its numerical properties in simulations calibrated to existing experiments.

Chapter 2 studies how to allocate treatments on a network, using information from an existing (quasi)-experiment. We introduce a method that maximizes the sample analog of social welfare whenspillovers occur. We construct semi-parametric welfare estimators and cast the optimization problem into a mixed-integer linear program. We derive guarantees on the regret and illustrate the method's advantage when targeting information on social networks.

Chapter 3 studies the problem of allocating treatments when policymakers have preferences for non-discriminatory actions. We adopt the non-maleficence perspective of ``first do no harm": we select the fairest allocation withinthe Pareto frontier. We derive off-the-shelf optimization procedures and regret bounds on the unfairness of the estimated policy function. We illustrate our method using an application from education economics.

Chapter 4 focuses on inference in the presence of multiple hypothesis testing. The use of multiple hypothesis testing adjustments varies widely in applied economic research, without consensus on when and how it should be done. We provide a game-theoretic foundation for this practice. We show that adjustments with many interventions must depend on the nature of scale economies in the research production function and on economic interactions between policy decisions. When research examines multiple outcomes, our framework motivates aggregating outcomes into sufficient statistics for policy-making.