Christian Gourieroux, Alain Monfort; On the Problem of Missing Data in Linear Models, The Review of Economic Studies, Volume 48, Issue 4, 1 October 1981, Pages

[This paper is concerned with linear models. In this context it gives a necessary and sufficient condition for equality of two generalized least squares estimators of any subset of the parameters. The result is applied to many kinds of problems: the Frisch-Waugh problem, equality of OLS and GLS estimators, equality of the single equation GLS and the overall GLS estimators in seemingly unrelated regressions, equality of partial and overall 3SLS, lineartransformation of a linear model, superfluous observations, and mixed estimation.]

The purpose of the paper is to introduce, in a discrete-time no-arbitrage pricing context, a bridge between the historical and the risk-neutral state vector dynamics which is wider than the one implied by a classical exponential-affine stochastic discount factor (SDF) and to preserve, at the same time, the tractability and flexibility of the associated asset pricing model. This goal is achieved by introducing the notion of exponential-quadratic SDF or, equivalently, the notion of Second-Order Esscher Transform. The log-pricing kernel is specified as a quadratic function of the factor and the associated sources of risk are priced by means of possibly non-linear stochastic first-order and second-order risk-correction coefficients. Focusing on security market models, this approach is developed in the multivariate conditionally Gaussian framework and its usefulness is testified by the specification and calibration of what we name the Second-Order GARCH Option Pricing Model. The associated European Call option pricing formula generates a rich family of implied volatility smiles and skews able to match the typically observed ones.

This article presents an intensity-based model of euro-area sovereign spreads. To identify liquidity-pricing effects, we exploit the information contained in the spreads between bonds issued by a German agency (KfW) and their sovereign counterparts. KfW’s liabilities being guaranteed by the German government, these spreads are essentially liquidity-driven. Liquidity effects are found to account for a sizeable share of spreads’ fluctuations. After having filtered risk premiums out of the spreads, we estimate the physical default probabilities of eleven countries. Physical probabilities of default are lower than risk-neutral ones, consistently with the existence of a nondiversifiable euro-area sovereign credit risk.

The recent macro-finance literature does not agree either about the empirical properties of the expectation part and of the term premium on long-term bonds or about the importance or even the direction of the relationship between the term premium and future economic activity. This paper proposes a two-step approach to handle both problems. First, in a VAR setting, we extract a reliable measure of the term premium by means of averaging estimator techniques aiming at optimally solving prediction problems when highly persistent processes are present and, thus, providing a so called Near-Cointegrated VAR(p) approach. Second, we analyze the dynamic response of GDP to shocks to the term premium by using the New Information Response Function concept. As far as the first problem is concerned, we find that the NCVAR-based term premium measure is rather stable and counter-cyclical, as suggested by interest rates survey-based estimation of yield curve models and by its risk compensation role. Regarding the second problem, we find that an increase in the long-term spread caused by the term premium induces two effects on future economic activity: the impact is negative for short horizons (less than 1year), whereas it is positive for longer ones.

We define a disastrous default as the default of a systemic entity. Such an event is expected to have a negative effect on the economy and to be contagious. Bringing macroeconomic structure to a no-arbitrage asset-pricing framework, we exploit prices of disaster-exposed assets (credit and equity derivatives) to extract information on (i) the expected influence of a disastrous default on consumption and (ii) the probability of a financial meltdown. Using European data, we find that the returns of disaster-exposed assets are consistent with a systemic default being followed by a 2% decrease in consumption. The recessionary influence of disastrous defaults implies that financial instruments whose payoffs are exposed to such credit events carry substantial risk premiums. We also produce systemic risk indicators based on the probability of observing a certain number of systemic defaults or a sharp drop of consumption.

The basic assumption of a structural vector autoregressive moving average (SVARMA) model is that it is driven by a white noise whose components are uncorrelated or independent and can be interpreted as economic shocks, called “structural” shocks. When the errors are Gaussian, independence is equivalent to non-correlation and these models face two identification issues. The first identification problem is “static” and is due to the fact that there is an infinite number of linear transformations of a given random vector making its components uncorrelated. The second identification problem is “dynamic” and is a consequence of the fact that, even if a SVARMA admits a non-invertible moving average (MA) matrix polynomial, it may feature the same second-order dynamic properties as a VARMA process in which the MA matrix polynomials are invertible (the fundamental representation). The aim of this article is to explain that these difficulties are mainly due to the Gaussian assumption, and that both identification challenges are solved in a non-Gaussian framework if the structural shocks are assumed to be instantaneously and serially independent. We develop new parametric and semi-parametric estimation methods that accommodate non-fundamentalness in the MA dynamics. The functioning and performances of these methods are illustrated by applications conducted on both simulated and real data.

This paper considers the problem of testing statistical hypotheses in linear regression models with inequality constraints on the regression coefficients. The Kuhn-Tucker multiplier test statistic is defined and its relationships with the likelihood ratio test and the Wald test are examined. It is shown, in particular, that these relationships are the same as in the equality constrained case. It is emphasized, however, that their common asymptotic distribution is a mixture of chi-square distributions under the null hypothesis.