Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| < ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [sum ]k=0∞ |ψk| < ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations εk form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations εk is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.

The article reviews the early history of the Cowles Commission (CC), its close and intertwined relations with the Econometric Society (ES), and the influence and guidance of Ragnar Frisch. It provides detail on three rounds of choosing a research director for CC in 1937–38, in 1939 and, particularly, at the end of 1942. Haavelmo’s work in the early 1940s came to play a major role for the econometric research at the Cowles Commission under Jacob Marschak as research director 1943–48. The article points to the importance of Abraham Wald and Jacob Marschak for the success of Haavelmo’s venture and its influence and tells the story of how it came about that Haavelmo’s ideas were adopted, applied, and disseminated by the Cowles Commission. Thus the mention of Trygve Haavelmo in the title is referring also to his econometric ideas. The ideas themselves and their further evolvement at the CC have been a dominating theme in the history of econometrics literature, e.g., Hildreth (1986), Epstein (1987), Morgan (1990), Qin (1993), and Christ (1994). The article discusses the recruitment, the inner workings, and various other concerns of the Cowles econometricians, from Marxism to Black Magic. It recounts at some length the efforts made by Marschak to recruit Abraham Wald to the University of Chicago and the Cowles Commission. The article can be read as a sequel to Bjerkholt (2005, 2007).

Sir Richard Stone, knighted in 1978 and Nobel Laureate in Economics in 1984, is one of the pioneering architects of national income and social accounts, and is one of the few economists of his generation to have faced the challenge of economics as a science by combining theory and measurement within a cohesive framework. He was awarded the Nobel Prize in Economics for his “fundamental contributions to the development of national accounts,” but he has made equally significant contributions to the empirical analysis of consumer behavior. His work on the “Growth Project” has also been instrumental in the development of appropriate econometric methodology for the construction and the analysis of large disaggregated macroeconometric models.

Throughout his long and productive career, stretching over more than half a century, Stone has been an inspiration to applied econometricians all over the world. His influence goes well beyond his written work. He has made a lasting impact on the large number of (now prominent) economists and statisticians who visited the Department of Applied Economics when he was its Director. He is a scientist, a scholar, and above all, a gentleman. He gives generously of himself and is always willing to help the cause of applied econometrics. He has been a Fellow of King's College, Cambridge since 1945 and has served as the President of the Econometric Society (in 1955) and the President of the Royal Economic Society (during 1978–1980). In the interview that follows, Richard Stone gives us a delightful account of his time as a student at Westminster School, his early introduction to economics at Cambridge University, and he shares with us his memories and thoughts on a long and productive career. The interview was conducted in Stones' magnificent private library in Cambridge, and I hope that readers enjoy reading the interview as much as I enjoyed recording it.

Further details of Richard Stone's biography and research activities can be found in:

Deaton, A. Stone, John Richard Nicholas. In J. Eatwell, M. Milgate and P. Newman (eds.), The New Palgrave: A Dictionary of Economics, Vol. 4, pp. 509–512. London: Macmillan, 1987.

Stone, J.R.N. An autobiographical sketch. In Les Prix Nobel 1984. Stockholm: Almquist and Wicksell International, 1985.

Closed forms for the distribution of some conventional statistics are given as a prelude to deriving their asymptotic power functions as unit root tests. In the process, an important distinction is drawn between two classes of statistics: one which relies on deterministic normalizations and the other which uses stochastic normalizations. When the data follow a driftless autoregression, a t test (which belongs to the second class) for a unit root is found to perform better than the other tests in small to moderate effective samples.

An asymptotic theory is developed for a weakly identified cointegrating regression model in which the regressor is a nonlinear transformation of an integrated process. Weak identification arises from the presence of a loading coefficient for the nonlinear function that may be close to zero. In that case, standard nonlinear cointegrating limit theory does not provide good approximations to the finite-sample distributions of nonlinear least squares estimators, resulting in potentially misleading inference. A new local limit theory is developed that approximates the finite-sample distributions of the estimators uniformly well irrespective of the strength of the identification. An important technical component of this theory involves new results showing the uniform weak convergence of sample covariances involving nonlinear functions to mixed normal and stochastic integral limits. Based on these asymptotics, we construct confidence intervals for the loading coefficient and the nonlinear transformation parameter and show that these confidence intervals have correct asymptotic size. As in other cases of nonlinear estimation with integrated processes and unlike stationary process asymptotics, the properties of the nonlinear transformations affect the asymptotics and, in particular, give rise to parameter dependent rates of convergence and differences between the limit results for integrable and asymptotically homogeneous functions.

We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of xt = Δ−dut, where is the fractional integration parameter and ut is weakly dependent. The classical condition is existence of q ≥ 2 and moments of the innovation sequence. When d is close to this moment condition is very strong. Our main result is to show that when and under some relatively weak conditions on ut, the existence of moments is in fact necessary for the FCLT for fractionally integrated processes and that moments are necessary for more general fractional processes. Davidson and de Jong (2000, Econometric Theory 16, 643–666) presented a fractional FCLT where only q > 2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient and hence that their result is incorrect.

In this paper we provide some weak convergence results for sample statistics of the product of a variable with its kth-order lag. We assume the variable is a stationary vector that can be represented by linear process, and the lag length k is allowed to be a function of the sample size. Employing the Beveridge–Nelson decomposition, we derive a new functional central limit theorem for this situation and establish related stochastic integral convergence results. We then consider the behavior of associated long-run variance estimators and also extend our analysis to the case where the sample statistics are based on regression residuals. We illustrate the potential range of application of these techniques in the context of (i) testing for I(0) versus I(1) behavior and (ii) estimation and testing in a heteroskedastically cointegrated regression model.We thank the co-editor and the referees for helpful comments on earlier drafts.

To obtain consistency results for nonparametric estimators based on stochastic processes relevant in econometrics, we introduce the notions of Hilbert space-valued Lp mixingales and near-epoch dependent arrays, and we prove weak and strong laws of large numbers by using a new exponential inequality for Hilbert (H) space-valued martingale difference arrays. We follow Andrews (1988, Econometric Theory 4, 458–467), Hansen (1991, Econometric Theory 7, 213–221; 1992, Econometric Theory 8, 421–422), Davidson (1993, Statistics and Probability Letters 16,301–304), and de Jong (1995, Econometric Theory 11, 347–358), extending results for H = R and improving memory conditions in certain instances. We give as examples consistency results for series and kernel estimators.

In this article, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of Lp functionals of kernel-type estimators (1 ≤ p < ∞) and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on the nonparametric bootstrap applied to kernel-based test statistics, with an option of estimating “contact sets.” We provide regularity conditions under which the bootstrap test is asymptotically valid uniformly over a large class of distributions, including cases where the limiting distribution of the test statistic is degenerate. Our bootstrap test is shown to exhibit good power properties in Monte Carlo experiments, and we provide a general form of the local power function. As an illustration, we consider testing implications from auction theory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We supplement this example with the second empirical illustration in the context of wage inequality.

This paper is concerned with the progress of Trygve Haavelmo's research and with his activities in general during his stay in the United States from June 1939 until the publication of his thesis “The Probability Approach in Econometrics” (Haavelmo, 1944, Econometrica 12, Suppl. 1–118) in July 1944. His original intention had been to stay in the United States only until the end of 1940, but the outbreak of World War II and the German occupation of Norway left him stranded. His “Theory and Measurement” treatise (Haavelmo, 1941, “On the Theory and Measurement of Economic Relations,” hectograph), the first version of “The Probability Approach,” was completed by the middle of 1941. From 1942 Haavelmo worked in New York for the Norwegian government in exile and was called upon to present his ideas to econometricians. Throughout his time in the United States he argued that probability should be accepted as an integral part of economic theory and as a basis for verification in economics. This paper considers how much of Haavelmo's approach was the result of his prewar experiences and how much the result of his time in the United States. It elaborates on his contact with Jerzy Neyman, Abraham Wald, and Jakob Marschak and the circumstances leading to the publication of “The Probability Approach.” Haavelmo's activities have been tracked through letters, seminar notes, and a reconstruction of his itinerary. The paper is a sequel to Bjerkholt (2005, Econometric Theory 21, 491–533).The presentation owes very much indeed to the advice of three anonymous referees and to the editor Peter Phillips, as the result of which it was restructured, reformulated, and much improved. I am sincerely grateful to all of these individuals for their generous and constructive advice. The source material for the paper is to a large extent correspondence and documents from the Haavelmo Archive at the Department of Economics, University of Oslo, as organized by Tore Thonstad. It also draws on material from the Frisch correspondence files at the National Library of Norway and from the Frisch Archive at the Department of Economics, University of Oslo. I am most grateful to Wendy Glickman of the Rockefeller Archive Center for providing me with copies of the monitoring sheets on Haavelmo. I owe my colleague Tore Schweder thanks for very helpful advice. I have furthermore benefited from personal communication with Ted Anderson, Kenneth J. Arrow, Leo Hurwicz, Eilev S. Jansen, Lawrence R. Klein, J.J. Polak, and Paul A. Samuelson, for which I am most grateful. NBER's Claudia Goldin, Chris Nagorski, and Robert Lipsey helped me sort out some puzzles. Thanks are also due to Lesney Levene for invaluable linguistic advice and to Inger Bjerkodden for encouragement all along. The responsibility for all remaining errors (surely, a nonempty set) is entirely my own.

We analyze estimators and tests for a general class of vector error correction models that allows for asymmetric and nonlinear error correction. For a given number of cointegration relationships, general hypothesis testing is considered, where testing for linearity is of particular interest. Under the null of linearity, parameters of nonlinear components vanish, leading to a nonstandard testing problem. We apply so-called sup-tests to resolve this issue, which requires development of new(uniform) functional central limit theory and results for convergence of stochastic integrals. We provide a full asymptotic theory for estimators and test statistics. The derived asymptotic results prove to be nonstandard compared to results found elsewhere in the literature due to the impact of the estimated cointegration relations. This complicates implementation of tests motivating the introduction of bootstrap versions that are simple to compute. A simulation study shows that the finite-sample properties of the bootstrapped tests are satisfactory with good size and power properties for reasonable sample sizes.

Forecasts from a univariate autoregressive model estimated by OLS are unbiased, irrespective of whether the model fitted has the correct order; this property only requires symmetry of the distribution of the innovations. In this paper, this result is generalized to vector autoregressions and a wide class of multivariate stochastic processes (which include Gaussian stationary multivariate stochastic processes) is described for which unbiasedness of predictions holds: specifically, if a vector autoregression of arbitrary finite order is fitted to a sample from any process in this class, the fitted model will produce unbiased forecasts, in the sense that the prediction errors have distributions symmetric about zero. Different numbers of lags may be used for each variable in each autoregression and variables may even be missing, without unbiasedness being affected. This property is exact in finite samples. Similarly, the residuals from the same autoregressions have distributions symmetric about zero.

There are many instances in the statistical literature in which inference is based on a normalized quadratic form in a standard normal vector, normalized by the squared length of that vector. Examples include both test statistics (the Durbin–Watson statistic) and estimators (serial correlation coefficients). Although much studied, no general closed-form expression for the density function of such a statistic is known. This paper gives general formulae for the density in each open interval between the characteristic roots of the matrix involved. Results are given for the case of distinct roots, which need not be assumed positive, and when the roots occur with multiplicities greater than one. Starting from a representation of the density as a surface integral over an (n − 2)-dimensional hyperplane, the density is expressed in terms of top-order zonal polynomials involving difference quotients of the characteristic roots of the matrix in the numerator quadratic form.

This paper provides sufficient conditions for the nonparametric identification of the regression function $m\left( \cdot \right)$ in a regression model with an endogenous regressor x and an instrumental variable z. It has been shown that the identification of the regression function from the conditional expectation of the dependent variable on the instrument relies on the completeness of the distribution of the endogenous regressor conditional on the instrument, i.e., $f\left( {x|z} \right)$. We show that (1) if the deviation of the conditional density $f\left( {x|{z_k}} \right)$ from a known complete sequence of functions is less than a sequence of values determined by the complete sequence in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$, then $f\left( {x|z} \right)$ itself is complete, and (2) if the conditional density $f\left( {x|z} \right)$ can form a linearly independent sequence $\{ f( \cdot |{z_k}):k = 1,2, \ldots \}$ in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$ and its relative deviation from a known complete sequence of functions under some norm is finite then $f\left( {x|z} \right)$ itself is complete. We use these general results to provide specific sufficient conditions for completeness in three different specifications of the relationship between the endogenous regressor x and the instrumental variable $z.$

Commonly used tests to assess evidence for the absence of autocorrelation in a univariate time series or serial cross-correlation between time series rely on procedures whose validity holds for i.i.d. data. When the series are not i.i.d., the size of correlogram and cumulative Ljung–Box tests can be significantly distorted. This paper adapts standard correlogram and portmanteau tests to accommodate hidden dependence and nonstationarities involving heteroskedasticity, thereby uncoupling these tests from limiting assumptions that reduce their applicability in empirical work. To enhance the Ljung–Box test for non-i.i.d. data, a new cumulative test is introduced. Asymptotic size of these tests is unaffected by hidden dependence and heteroskedasticity in the series. Related extensions are provided for testing cross-correlation at various lags in bivariate time series. Tests for the i.i.d. property of a time series are also developed. An extensive Monte Carlo study confirms good performance in both size and power for the new tests. Applications to real data reveal that standard tests frequently produce spurious evidence of serial correlation.

The exact discrete model satisfied by equispaced data generated by a linear stochastic differential equations system is derived by a method that does not imply restrictions on observed discrete data per se. The method involves integrating the solution of the continuous time model in state space form and a nonstandard change in the order of three types of integration, facilitating the representation of the exact discrete model as an asymptotically time-invariant vector autoregressive moving average model. The method applying to the state space form is general and is illustrated using the prototypical higher order model for mixed stock and flow data discussed by Bergstrom (1986, Econometric Theory 2, 350–373).

Mundlak (1978, Econometrica 46, 69–85) showed that the fixed effects estimator can be obtained as generalized least squares (GLS) for a panel regression model where the individual effects are random but are all hopelessly correlated with the regressors. This result was obtained by partitioned inversion after substituting the reduced form expression for the individual effects as a function of the means of all the regressors. This note shows that Mundlak's result can be obtained using system estimation without using partitioned inversion. System estimation has proved useful for deriving two-stage least squares (2SLS) and three-stage least squares (3SLS) counterparts for the random effects panel models by Baltagi (1981, Journal of Econometrics 17, 189–200). It also has been used for obtaining an alternative derivation of the Hausman tests that is robust to heteroskedasticity of unknown form (see Arellano, 1993, Journal of Econometrics 59, 87–97) and more recently, for obtaining generalized method of moments (GMM) estimators for dynamic panel models (see Arellano and Bover, 1995, Journal of Econometrics 68, 29–51; and Blundell and Bond, 1998, Journal of Econometrics 87, 115–143, to mention a few). We also show that a necessary and sufficient condition for ordinary least squares (OLS) to be equivalent to GLS is satisfied for this model.

Three important issues surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data; uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not, and the possible presence of nonstationary volatility in the data. Assuming homoskedasticity, Harvey, Leybourne, and Taylor (2011, Journal of Econometrics, forthcoming) propose decision rules based on a four-way union of rejections of quasi-differenced (QD) and ordinary least squares (OLS) detrended tests, both with and without a linear trend, to deal with the first two problems. In this paper we first discuss, again under homoskedasticity, how these union tests may be validly bootstrapped using the sieve bootstrap principle combined with either the independent and identically distributed (i.i.d.) or wild bootstrap resampling schemes. This serves to highlight the complications that arise when attempting to bootstrap the union tests. We then demonstrate that in the presence of nonstationary volatility the union test statistics have limit distributions that depend on the form of the volatility process, making tests based on the standard asymptotic critical values or, indeed, the i.i.d. bootstrap principle invalid. We show that wild bootstrap union tests are, however, asymptotically valid in the presence of nonstationary volatility. The wild bootstrap union tests therefore allow for a joint treatment of all three of the aforementioned issues in practice.