In regression analysis, some predictors might be unobservable, not observed, or ignored. These factors actually affect the response randomly. The observed data thus follows a conditional distribution when these factors are given. This phenomenon is called the distribution randomness. For such a working model, we propose an upper expectation regression and a two-step penalized maximum least squa...

This paper examines how the estimation results for a standard New Keynesian model with constant gain least squares learning is sensitive to the stance taken on agents beliefs at the beginning of the sample. The New Keynesian model is estimated under rational expectations and under learning with three di erent frameworks for how expectations are set at the beginning of the sample. The results sh...

This is also called the dot product and written X ·Y . The inner product of two vectors is a number, not another vector. In particular, we have the vital identity ‖X‖2 = 〈X , X〉 relating the inner product and norm. For added clarity, it is sometimes useful to write the inner product in Rn as 〈X , Y 〉Rn . Example: In R4 , if X = (1,2,−2,0) and Y = (−1,2,3,4) , then 〈X , Y 〉 = (1)(−1) + (2)(2)+(−...

We study the asymptotic behavior of the least squares estimators of the unknown parameters of bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the central limit theorem...

This paper continues the empirical research line started by Crepon et al. (1998) about the impact of research and development on innovation and innovation on productivity of firms. In this paper we estimate a structural model using Asymptotic Least Squares (ALS) which corrects for selectivity and simultaneity biases taking in consideration the particular characteristics of the available data. W...

A deterministic interpretation of the Kalman !ltering formulas is given, using the principle of least squares estimation. The observed signal and the to-be-estimated signal are modeled as being generated as outputs of a !nite-dimensional linear system driven by an input disturbance. Postulating that the observed signal is generated by the input disturbance that has minimal least squares norm le...

In the estimation of models with averaged data, weighted least squares is often used and recommended as a way of improving the e¢ciency of the estimator. However, if the size of the di¤erent groups is not conditionally independent of the regressand, consistent estimation may not be possible at all. It is argued that in the case of some leading examples of averaged data regression, consistent es...

4 Appendix S-1 4.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S-1 4.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S-4 4.3 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S-8 4.4 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

The first part of this paper contains an overview of the least squares method applied to various problems in ordinary and partial differential equations. In particular, we discuss various applications to the homogenization of transport equations, to the characterization of entropy solutions to scalar conservation laws and to the asymptotic behaviour of the action functional obtained through the...