## C32 Time-Series Models; Dynamic Quantile Regressions (Updated!)

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Search costs for lenders when evaluating potential borrowers are driven by the quality of the underwriting model and by access to data. Both have undergone radical change over the last years, due to the advent of big data and machine learning. For some, this holds the promise of inclusion and better access to finance. Invisible prime applicants perform better under AI than under traditional metrics. Broader data and more refined models help to detect them without triggering prohibitive costs. However, not all applicants profit to the same extent. Historic training data shape algorithms, biases distort results, and data as well as model quality are not always assured. Against this background, an intense debate over algorithmic discrimination has developed. This paper takes a first step towards developing principles of fair lending in the age of AI. It submits that there are fundamental difficulties in fitting algorithmic discrimination into the traditional regime of anti-discrimination laws. Received doctrine with its focus on causation is in many cases ill-equipped to deal with algorithmic decision-making under both, disparate treatment, and disparate impact doctrine. The paper concludes with a suggestion to reorient the discussion and with the attempt to outline contours of fair lending law in the age of AI.

Linear rational-expectations models (LREMs) are conventionally "forwardly" estimated as follows. Structural coefficients are restricted by economic restrictions in terms of deep parameters. For given deep parameters, structural equations are solved for "rational-expectations solution" (RES) equations that determine endogenous variables. For given vector autoregressive (VAR) equations that determine exogenous variables, RES equations reduce to reduced-form VAR equations for endogenous variables with exogenous variables (VARX). The combined endogenous-VARX and exogenous-VAR equations comprise the reduced-form overall VAR (OVAR) equations of all variables in a LREM. The sequence of specified, solved, and combined equations defines a mapping from deep parameters to OVAR coefficients that is used to forwardly estimate a LREM in terms of deep parameters. Forwardly-estimated deep parameters determine forwardly-estimated RES equations that Lucas (1976) advocated for making policy predictions in his critique of policy predictions made with reduced-form equations.
Sims (1980) called economic identifying restrictions on deep parameters of forwardly-estimated LREMs "incredible", because he considered in-sample fits of forwardly-estimated OVAR equations inadequate and out-of-sample policy predictions of forwardly-estimated RES equations inaccurate. Sims (1980, 1986) instead advocated directly estimating OVAR equations restricted by statistical shrinkage restrictions and directly using the directly-estimated OVAR equations to make policy predictions. However, if assumed or predicted out-of-sample policy variables in directly-made policy predictions differ significantly from in-sample values, then, the out-of-sample policy predictions won't satisfy Lucas's critique.
If directly-estimated OVAR equations are reduced-form equations of underlying RES and LREM-structural equations, then, identification 2 derived in the paper can linearly "inversely" estimate the underlying RES equations from the directly-estimated OVAR equations and the inversely-estimated RES equations can be used to make policy predictions that satisfy Lucas's critique. If Sims considered directly-estimated OVAR equations to fit in-sample data adequately (credibly) and their inversely-estimated RES equations to make accurate (credible) out-of-sample policy predictions, then, he should consider the inversely-estimated RES equations to be credible. Thus, inversely-estimated RES equations by identification 2 can reconcile Lucas's advocacy for making policy predictions with RES equations and Sims's advocacy for directly estimating OVAR equations.
The paper also derives identification 1 of structural coefficients from RES coefficients that contributes mainly by showing that directly estimated reduced-form OVAR equations can have underlying LREM-structural equations.

Several recent studies have expressed concern that the Haar prior typically imposed in estimating sign-identi.ed VAR models may be unintentionally informative about the implied prior for the structural impulse responses. This question is indeed important, but we show that the tools that have been used in the literature to illustrate this potential problem are invalid. Speci.cally, we show that it does not make sense from a Bayesian point of view to characterize the impulse response prior based on the distribution of the impulse responses conditional on the maximum likelihood estimator of the reduced-form parameters, since the the prior does not, in general, depend on the data. We illustrate that this approach tends to produce highly misleading estimates of the impulse response priors. We formally derive the correct impulse response prior distribution and show that there is no evidence that typical sign-identi.ed VAR models estimated using conventional priors tend to imply unintentionally informative priors for the impulse response vector or that the corre- sponding posterior is dominated by the prior. Our evidence suggests that concerns about the Haar prior for the rotation matrix have been greatly overstated and that alternative estimation methods are not required in typical applications. Finally, we demonstrate that the alternative Bayesian approach to estimating sign-identi.ed VAR models proposed by Baumeister and Hamilton (2015) su¤ers from exactly the same conceptual shortcoming as the conventional approach. We illustrate that this alternative approach may imply highly economically implausible impulse response priors.

Analysing causality among oil prices and, in general, among financial and economic variables is of central relevance in applied economics studies. The recent contribution of Lu et al. (2014) proposes a novel test for causality— the DCC-MGARCH Hong test. We show that the critical values of the test statistic must be evaluated through simulations, thereby challenging the evidence in papers adopting the DCC-MGARCH Hong test. We also note that rolling Hong tests represent a more viable solution in the presence of short-lived causality periods.

Empirical estimates of equilibrium real interest rates are so far mostly limited to advanced economies, since no statistical procedure suitable for a large set of countries is available. This is surprising, as equilibrium rates have strong policy implications in emerging markets and developing economies as well; current estimates of the global equilibrium rate rely on only a few countries; and estimates for a more diverse set of countries can improve understanding of the drivers. The authors propose a model and estimation strategy that decompose ex ante real interest rates into a permanent and transitory component even with short samples and high volatility. This is done with an unobserved component local level stochastic volatility model, which is used to estimate equilibrium rates for 50 countries with Bayesian methods.
Equilibrium rates were lower in emerging markets and developing economies than in advanced economies in the 1980s, similar in the 1990s, and have been higher since 2000. In line with economic integration and rising global capital markets, synchronization has been rising over time and is higher among advanced economies. Equilibrium rates of countries with stronger trade linkages and similar demographic and economic trends are more synchronized.

We derive the Bayes estimator of vectors of structural VAR impulse responses under a range of alternative loss functions. We also derive joint credible regions for vectors of impulse responses as the lowest posterior risk region under the same loss functions. We show that conventional impulse response estimators such as the posterior median response function or the posterior mean response function are not in general the Bayes estimator of the impulse response vector obtained by stacking the impulse responses of interest. We show that such pointwise estimators may imply response function shapes that are incompatible with any possible parameterization of the underlying model. Moreover, conventional pointwise quantile error bands are not a valid measure of the estimation uncertainty about the impulse response vector because they ignore the mutual dependence of the responses. In practice, they tend to understate substantially the estimation uncertainty about the impulse response vector.

We analyze cyclical co-movement in credit, house prices, equity prices, and longterm interest rates across 17 advanced economies. Using a time-varying multi-level dynamic factor model and more than 130 years of data, we analyze the dynamics of co-movement at different levels of aggregation and compare recent developments to earlier episodes such as the early era of financial globalization from 1880 to 1913 and the Great Depression. We find that joint global dynamics across various financial quantities and prices as well as variable-specific global co-movements are important to explain fluctuations in the data. From a historical perspective, global co-movement in financial variables is not a new phenomenon, but its importance has increased for some variables since the 1980s. For equity prices, global cycles play currently a historically unprecedented role, explaining more than half of the fluctuations in the data. Global cycles in credit and housing have become much more pronounced and longer, but their importance in explaining dynamics has only increased for some economies including the US, the UK and Nordic European countries. We also include GDP in the analysis and find an increasing role for a global business cycle.

Extending the data set used in Beyer (2009) to 2017, we estimate I(1) and I(2) money demand models for euro area M3. After including two broken trends and a few dummies to account for shifts in the variables following the global financial crisis and the ECB's non-standard monetary policy measures, we find that the money demand and the real wealth relations identified in Beyer (2009) have remained remarkably stable throughout the extended sample period. Testing for price homogeneity in the I(2) model we find that the nominal-to-real transformation is not rejected for the money relation whereas the wealth relation cannot be expressed in real terms.

The authors relax the standard assumption in the dynamic stochastic general equilibrium (DSGE) literature that exogenous processes are governed by AR(1) processes and estimate ARMA (p,q) orders and parameters of exogenous processes. Methodologically, they contribute to the Bayesian DSGE literature by using Reversible Jump Markov Chain Monte Carlo (RJMCMC) to sample from the unknown ARMA orders and their associated parameter spaces of varying dimensions.
In estimating the technology process in the neoclassical growth model using post war US GDP data, they cast considerable doubt on the standard AR(1) assumption in favor of higher order processes. They find that the posterior concentrates density on hump-shaped impulse responses for all endogenous variables, consistent with alternative empirical estimates and the rigidities behind many richer structural models. Sampling from noninvertible MA representations, a negative response of hours to a positive technology shock is contained within the posterior credible set. While the posterior contains significant uncertainty regarding the exact order, the results are insensitive to the choice of data filter; this contrasts with the authors’ ARMA estimates of GDP itself, which vary significantly depending on the choice of HP or first difference filter.