A crucial input into causal inference is the imputed counterfactual outcome. Imputation error can arise because of sampling uncertainty from estimating the prediction model using the untreated observations, or from out-of-sample information not captured by the model. While the literature has focused on sampling uncertainty, it vanishes with the sample size. Often overlooked is the possibility that the out-of-sample error can be informative about the missing counterfactual outcome if it is mutually or serially correlated. Motivated by the best linear unbiased predictor (\blup) of \citet{goldberger:62} in a time series setting, we propose an improved predictor of potential outcome when the errors are correlated. The proposed \pup\; is practical as it is not restricted to linear models, can be used with consistent estimators already developed, and improves mean-squared error for a large class of strong mixing error processes. Ignoring predictability in the errors can distort conditional inference. However, the precise impact will depend on the choice of estimator as well as the realized values of the residuals.

Monthly and weekly economic indicators are often taken to be the largest common factor estimated from high and low frequency data, either separately or jointly. To incorporate mixed frequency information without directly modeling them, we target a low frequency diffusion index that is already available, and treat high frequency values as missing. We impute these values using multiple factors estimated from the high frequency data. In the empirical examples considered, static matrix completion that does not account for serial correlation in the idiosyncratic errors yields imprecise estimates of the missing values irrespective of how the factors are estimated. Single equation and systems-based dynamic procedures that account for serial correlation yield imputed values that are closer to the observed low frequency ones. This is the case in the counterfactual exercise that imputes the monthly values of consumer sentiment series before 1978 when the data was released only on a quarterly basis. This is also the case for a weekly version of the CFNAI index of economic activity that is imputed using seasonally unadjusted data. The imputed series reveals episodes of increased variability of weekly economic information that are masked by the monthly data, notably around the 2014-15 collapse in oil prices.

Economists are blessed with a wealth of data for analysis, but more often than not, values in some entries of the data matrix are missing. Various methods have been proposed to handle missing observations in a few variables. We exploit the factor structure in panel data of large dimensions. Our \textsc{tall-project} algorithm first estimates the factors from a \textsc{tall} block in which data for all rows are observed, and projections of variable specific length are then used to estimate the factor loadings. A missing value is imputed as the estimated common component which we show is consistent and asymptotically normal without further iteration. Implications for using imputed data in factor augmented regressions are then discussed. To compensate for the downward bias in covariance matrices created by an omitted noise when the data point is not observed, we overlay the imputed data with re-sampled idiosyncratic residuals many times and use the average of the covariances to estimate the parameters of interest. Simulations show that the procedures have desirable finite sample properties.

We propose a method to conduct uniform inference for the (optimal) value function, that is, the function that results from optimizing an objective function marginally over one of its arguments. Marginal optimization is not Hadamard differentiable (that is, compactly differentiable) as a map between the spaces of objective and value functions, which is problematic because standard inference methods for nonlinear maps usually rely on Hadamard differentiability. However, we show that the map from objective function to an Lp functional of a value function, for 1≤p≤∞, are Hadamard directionally differentiable. As a result, we establish consistency and weak convergence of nonparametric plug-in estimates of Cramér–von Mises and Kolmogorov–Smirnov test statistics applied to value functions. For practical inference, we develop detailed resampling techniques that combine a bootstrap procedure with estimates of the directional derivatives. In addition, we establish local and uniform size control of one-sided tests which use the resampling procedure. Monte Carlo simulations assess the finite-sample properties of the proposed methods and show accurate empirical size and nontrivial power of the procedures. Finally, we apply our methods to the evaluation of a job training program using bounds for the distribution function of treatment effects.

Pervasive cross-section dependence is increasingly recognized as a characteristic of economic data and the approximate factor model provides a useful framework for analysis. Assuming a strong factor structure where Λ0′Λ0/Nα is positive definite in the limit when α=1, early work established convergence of the principal component estimates of the factors and loadings up to a rotation matrix. This paper shows that the estimates are still consistent and asymptotically normal when α∈(0,1] albeit at slower rates and under additional assumptions on the sample size. The results hold whether α is constant or varies across factor loadings. The framework developed for heterogeneous loadings and the simplified proofs that can be also used in strong factor analysis are of independent interest.

This paper provides three results for SVARs under the assumption that the primitive shocks are mutually independent. First, a framework is proposed to accommodate a disaster-type variable with infinite variance into a SVAR. We show that the least squares estimates of the SVAR are consistent but have non-standard asymptotics. Second, the disaster shock is identified as the component with the largest kurtosis. An estimator that is robust to infinite variance is used to recover the mutually independent components. Third, an independence test on the residuals pre-whitened by the Choleski decomposition is proposed to test the restrictions imposed on a SVAR. The test can be applied whether the data have fat or thin tails, and to over as well as exactly identified models. Three applications are considered. In the first, the independence test is used to shed light on the conflicting evidence regarding the role of uncertainty in economic fluctuations. In the second, disaster shocks are shown to have short term economic impact arising mostly from feedback dynamics. The third uses the framework to study the dynamic effects of economic shocks post-covid.

Researchers may perform regressions using a sketch of data of size m instead of the full sample of size n for a variety of reasons. This paper considers the case when the regression errors do not have constant variance and heteroskedasticity robust standard errors would normally be needed for test statistics to provide accurate inference. We show that estimates using data sketched by random projections will behave 'as if' the errors were homoskedastic. Estimation by random sampling would not have this property. The result arises because the sketched estimates in the case of random projections can be expressed as degenerate U-statistics, and under certain conditions, these statistics are asymptotically normal with homoskedastic variance. We verify that the conditions hold not only in the case of least squares regression when the covariates are exogenous, but also in instrumental variables estimation when the covariates are endogenous. The result implies that inference can be simpler than the full sample case if the sketching scheme is appropriately chosen.

Beliefs are important determinants of an individual's choices and economic outcomes, so understanding how they differ across individuals is of considerable interest. Researchers often rely on surveys that report individual expectations as qualitative data. We propose using a Bayesian hierarchical latent class model to summarize and interpret observed heterogeneity in categorical expectations data. We show that the statistical model corresponds to an economic structural model of information acquisition, which guides interpretation and estimation of the model parameters. An algorithm based on stochastic optimization is proposed to estimate a model for repeated surveys when beliefs follow a dynamic structure and conjugate priors are not appropriate. Guidance on selecting the number of belief types is also provided. Two examples are considered. The first shows that there is information in the Michigan survey responses beyond the consumer sentiment index that is officially published. The second shows that belief types constructed from survey responses can be used in a subsequent analysis to estimate heterogeneous returns to education.

High dimensional predictive regressions are useful in wide range of applications. However, the theory is mainly developed assuming that the model is stationary with time invariant parameters. This is at odds with the prevalent evidence for parameter instability in economic time series, but theories for parameter instability are mainly developed for models with a small number of covariates. In this paper, we present two $L_2$ boosting algorithms for estimating high dimensional models in which the coefficients are modeled as functions evolving smoothly over time and the predictors are locally stationary. The first method uses componentwise local constant estimators as base learner, while the second relies on componentwise local linear estimators. We establish consistency of both methods, and address the practical issues of choosing the bandwidth for the base learners and the number of boosting iterations. In an extensive application to macroeconomic forecasting with many potential predictors, we find that the benefits to modeling time variation are substantial and they increase with the forecast horizon. Furthermore, the timing of the benefits suggests that the Great Moderation is associated with substantial instability in the conditional mean of various economic series.

Using monthly data on costly natural disasters affecting the United States over the last 40 years, we estimate 2 time series models and use them to generate predictions about the impact of COVID-19. We find that while our models yield reasonable estimates of the impact on industrial production and the number of scheduled flight departures, they underestimate the unprecedented changes in the labor market.

Uncertainty about the future rises in recessions. But is uncertainty a source of business cycles or an endogenous response to them, and does the type of uncertainty matter? We propose a novel SVAR identification strategy to address these questions via inequality constraints on the structural shocks. We find that sharply higher macroeconomic uncertainty in recessions is often an endogenous response to output shocks, while uncertainty about financial markets is a likely source of output fluctuations. (JEL D81, E23, E32, E44, G14)

This paper proposes an imputation procedure that uses the factors estimated from a tall block along with the re-rotated loadings estimated from a wide block to impute missing values in a panel of data. Assuming that a strong factor structure holds for the full panel of data and its sub-blocks, it is shown that the common component can be consistently estimated at four different rates of convergence without requiring regularization or iteration. An asymptotic analysis of the estimation error is obtained. An application of our analysis is estimation of counterfactuals when potential outcomes have a factor structure. We study the estimation of average and individual treatment effects on the treated and establish a normal distribution theory that can be useful for hypothesis testing.

In this paper we present and describe a large quarterly frequency, macroeconomic database.The data provided are closely modeled to that used in Stock and Watson (2012a).As in our previous work on FRED-MD, our goal is simply to provide a publicly available source of macroeconomic "big data" that is updated in real time using the FRED database.We show that factors extracted from this data set exhibit similar behavior to those extracted from the original Stock and Watson data set.The dominant factors are shown to be insensitive to outliers, but outliers do affect the relative influence of the series as indicated by leverage scores.We then investigate the role unit root tests play in the choice of transformation codes with an emphasis on identifying instances in which the unit root-based codes differ from those already used in the literature.Finally, we show that factors extracted from our data set are useful for forecasting a range of macroeconomic series and that the choice of transformation codes can contribute substantially to the accuracy of these forecasts.

The coronavirus is a global event of historical proportions and just a few months changed the time series properties of the data in ways that make many pre-covid forecasting models inadequate. It also creates a new problem for estimation of economic factors and dynamic causal effects because the variations around the outbreak can be interpreted as outliers, as shifts to the distribution of existing shocks, or as addition of new shocks. I take the latter view and use covid indicators as controls to 'de-covid' the data prior to estimation. I find that economic uncertainty remains high at the end of 2020 even though real economic activity has recovered and covid uncertainty has receded. Dynamic responses of variables to shocks in a VAR similar in magnitude and shape to the ones identified before 2020 can be recovered by directly or indirectly modeling covid and treating it as exogenous. These responses to economic shocks are distinctly different from those to a covid shock which are much larger but shorter lived. Disentangling the two types of shocks can be important in macroeconomic modeling post-covid.

Datasets that are terabytes in size are increasingly common, but computer bottlenecks often frustrate a complete analysis of the data. While more data are better than less, diminishing returns suggest that we may not need terabytes of data to estimate a parameter or test a hypothesis. But which rows of data should we analyze, and might an arbitrary subset of rows preserve the features of the original data? This paper reviews a line of work that is grounded in theoretical computer science and numerical linear algebra, and which finds that an algorithmically desirable sketch, which is a randomly chosen subset of the data, must preserve the eigenstructure of the data, a property known as a subspace embedding. Building on this work, we study how prediction and inference can be affected by data sketching within a linear regression setup. We show that the sketching error is small compared to the sample size effect which a researcher can control. As a sketch size that is algorithmically optimal may not be suitable for prediction and inference, we use statistical arguments to provide 'inference conscious' guides to the sketch size. When appropriately implemented, an estimator that pools over different sketches can be nearly as efficient as the infeasible one using the full sample.

Estimates of the approximate factor model are increasingly used in empirical work. Their theoretical properties, studied some twenty years ago, also laid the ground work for analysis on large dimensional panel data models with cross-section dependence. This paper presents simplified proofs for the estimates by using alternative rotation matrices, exploiting properties of low rank matrices, as well as the singular value decomposition of the data in addition to its covariance structure. These simplifications facilitate interpretation of results and provide a more friendly introduction to researchers new to the field. New results are provided to allow linear restrictions to be imposed on factor models.

When there is so much data that they become a computation burden, it is not uncommon to compute quantities of interest using a sketch of data of size $m$ instead of the full sample of size $n$. This paper investigates the implications for two-stage least squares (2SLS) estimation when the sketches are obtained by a computationally efficient method known as CountSketch. We obtain three results. First, we establish conditions under which given the full sample, a sketched 2SLS estimate can be arbitrarily close to the full-sample 2SLS estimate with high probability. Second, we give conditions under which the sketched 2SLS estimator converges in probability to the true parameter at a rate of $m^{-1/2}$ and is asymptotically normal. Third, we show that the asymptotic variance can be consistently estimated using the sketched sample and suggest methods for determining an inference-conscious sketch size $m$. The sketched 2SLS estimator is used to estimate returns to education.

It is known that the common factors in a large panel of data can be consistently estimated by the method of principal components, and principal components can be constructed by iterative least squares regressions. Replacing least squares with ridge regressions turns out to have the effect of removing the contribution of factors associated with small singular values from the common component. The method has been used in the machine learning literature to recover low-rank matrices. We study the procedure from the perspective of estimating an approximate factor model. Under the rank-constraint, the common component is estimated by the space spanned by factors whose singular values exceed a threshold. The desire for minimum rank and parsimony lead to a data-dependent penalty for selecting the number of factors. The new criterion is more conservative than the existing deterministic penalties and is appropriate when the nominal number of factors is inflated by the presence of weak factors or large measurement noise. We provide asymptotic results that can be used to test economic hypotheses.