The bias of the least squares estimator over interval constraints

We show that the interval constrained least squares estimator for a regression model is in general bias. The bias and some of its properties are given when the regression residuals are normally distributed.     

Finite sample bias of the least squares estimator in an AR(p) model: estimation,inference, simulation and examples

This paper shows that the first order bias of least squares estimators of the coefficients of an AR(p) model is important for ‘typical’ macroeconomic time series and proposes a simple to apply method of bias reduction. Biases in individual coefficients often cumulate in the sum with far-reaching consequences for the cumulative impulse response function. This function, being nonlinear in the underlying coefficients, is particularly sensitive to biases when, as is often the case, the shocks are long-lived. Simulations and examples demonstrate some of the magnitudes involved.     

Results on the bias and inconsistency of ordinary least squares for the linear probability model

This note formalizes bias and inconsistency results for ordinary least squares (OLS) on the linear probability model and provides sufficient conditions for unbiasedness and consistency to hold. The conditions suggest that a “trimming estimator” may reduce OLS bias.     

A note on the computation of inequality constrained least squares estimates

We derive the implications of the Kuhn-Tucker conditions for the inequality constrained least squares problem. A novel implication is that if each imposed constraint fails if individually relaxed, at least one will fail if any number are relaxed jointly.     

The non-monotonicity of the bias and mean squared error of the two stage least squares estimators of exogenous variable coefficients

Owen (1976) [Owen, A.D., 1976. A proof that both the bias and mean squared error of the two stage least squares estimator are monotonically non-increasing functions of sample size. Econometrica 44, 409–411.] has shown that in a two equation static simultaneous equation model both the bias and mean squared error of the two stage least squares estimator of the endogenous variable coefficient are monotonically non-increasing functions of the sample size. This paper shows that neither property carries over to the exogenous variable coefficient estimator.     

Monte Carlo studies on the effectiveness of the bootstrap bias reduction method on 2SLS estimates

If first moments exist, two stage least squares estimators are consistent although biased. In this paper several bias correction methods are compared including bootstrap two stage least squares, Nagar's k-class and jackknife estimators for both parametric and non-parametric cases. Monte Carlo experiments on several models investigate the non-large sample properties of these estimators. The results strongly favor the bootstrap procedure judged by the amount of bias reduction and comparative variances.     

Generalized least squares and maximum likelihood estimation of the logistic function for technology diffusion

Given a simple stochastic model of technology adoption, we derive a function for technological diffusion that is logistic in the deterministic part and has an error term based on the binomial distribution. We derive two estimators—a generalized least squares (GLS) estimator and a maximum likelihood (ML) estimator—which should be more efficient than the ordinary least squares (OLS) estimators typically used to estimate technological diffusion functions. We compare the two new estimators with OLS using Monte-Carlo techniques and find that under perfect specification, GLS and ML are equally efficient and both are more efficient than OLS. There was no evidence of bias in any of the estimators. We used the estimators on some example data and found evidence suggesting that under conditions of misspecification, the estimated variance-covariance of the ML estimator is badly biased. We verified the existence of the bias with a second Monte-Carlo experiment performed with a known misspecification. In the second experiment, GLS was the most efficient estimator, followed by ML, and OLS was least efficient. We conclude that the GLS estimator of choice.     

A new perspective on the issue of selection bias in randomized controlled field experiments

Many randomized controlled trials require participants to opt in. Such self-selection could introduce a potential bias, because only the most optimistic may participate. We revisit this prediction. We argue that in many situations, the experimental intervention is competing with alternative interventions participants could conduct themselves outside the experiment. Since participants have a chance of being assigned to the control group, participating has a direct opportunity cost, which is likely to be higher for optimists. We propose a model of self-selection and show that both pessimists and optimists may opt out of the experiment, leading to an ambiguous selection bias.     

On the equivalence of indirect inference and bootstrap bias correction for linear IV estimators

This note shows that two ways of simulation based bias correction–indirect inference and bootstrap bias correction–are equivalent for two-stage-least-squares, as well as k$k$-class estimators for the standard linear model with endogenous regressors.     

On the asymptotic bias and mean squared error of an improved estimator for coefficients in linear regression

An estimator for regression coefficients of Kadiyala (1984) is considered. It is proved that the estimator is asymptotically unbiased. The asymptotic weak mean squared error of the estimator is also derived and it is proved that, under certain conditions, the estimator dominates a general class of estimators given by Vinod and Ullah (1981).     

The coefficient of the whole is an average of the coefficients of its parts: A special case of restricted least squares

This paper provides a consistent estimate of the bound of the marginal effect of an unobserved right-hand side variable on the dependent variable when only the sum of that variable with a positively correlated variable is available.     

A direct test of the endogeneity of money: Implications for Gulf Cooperation Council (GCC) countries

This paper contributes to the ongoing discussion about the endogeneity of money supply by empirically investigating the GCC countries. We propose and implement a direct test of money supply endogeneity that depends on econometric specification of exogeneity which has not been used in the literature before. To be able to make comparisons with previous studies, we also conducted Granger Causality tests to analyze the causality relationship between bank credit and money supply. Both of the empirical studies provide empirical evidence for the endogeneity of money supply in GCC countries. The results of the paper have many significant monetary policy implications for the upcoming monetary unification of the GCC countries.     

The out-of-sample performance of an exact median-unbiased estimator for the near-unity AR(1) model

We analyse the forecasting performance of several strategies when estimating the near-unity AR(1) model. We focus on the Andrews’ (1993) exact median-unbiased estimator (BC), the OLS estimator and the driftless random walk (RW). We also explore two pairwise combinations between these strategies. We do this to investigate whether BC helps in reducing forecast errors. Via simulations, we find that BC forecasts typically outperform OLS forecasts. When BC is compared to the RW we obtain mixed results, favouring the latter while the persistence of the true process increases. Interestingly, we find that the combination of BC-RW performs well in a near-unity scheme.     

A comparison of city size distributions for China and India from 1950 to 2010

We examine the distributions of Chinese and Indian city sizes for seven decades (1950s to 2010s) using lognormal, Pareto, and general Pareto distributions. We ascertain which distribution fits the data and how the city size distributions change during these periods. The Chinese city size distribution is represented by lognormal in the early periods (1950–1990) and by Pareto in 2010, but is not characterized by Zipf, which could be attributed to Chinese government’s restrictions of migration from rural to urban areas and the one-child policy. In contrast, the Indian city size distribution transitions from lognormal in the earlier periods to Zipf in the later periods.     

Modelling structural change in the UK housing market: A comparison of alternative house price models

This paper develops a new approach to the modelling of house prices in the UK, with housing demand being conditioned directly on consumers' expenditure rather than the determinants of expenditure. Conditioning on consumption ensures that the permanent income measure used in determining the level of consumption is consistently reflected in housing demand. The effects of financial liberalisation on the relative consumption of housing and non-housing goods and services are captured using the average loan-value ratio for first-time buyers. We also allow for financial effects via the real user cost of home ownership. House prices are assumed to adjust so as to clear the housing market. The proposed model is found to have structurally stable parameters across the housing market downturn since 1990. Statistical comparisons with the more conventional models in use at HM Treasury and the Bank of England during the early 1990s provide additional evidence in favour of our proposed approach.     

Money and prices: U.S. Data 1869–1914 (A study with directed graphs)

This paper reconsiders empirical evidence on relationships among money, income, nominal prices, and wheat prices. Error correction and directed acyclic graphs are used to study both lagged and contemporaneous relations in late 19^th and early 20^th century U.S. data. We summarize evidence supporting the view that money was a causal actor in price movement in this period. In the long run (at a five year horizon), over twenty percent of the movement in price is explained by earlier movements in money supply; whereas, wheat price accounts for less than ten percent of this movement. There is also evidence that money supply was not exogenous, as it was determined, in contemporaneous time, by movements in the general price level and income. About forty percent of the variation in money is explained by current or lagged prices and income. There remains considerable uncertainty with respect to role of wheat prices in this period. Innovations in wheat price explain over twenty five percent of the uncertainty in real income at the five year forecast horizon – suggesting wheat price as either causal or proxying for more fundamental causal forces in the U.S. economy over our period of analysis. First version received: December 1999/Final version received: February 2001     

The maximum number of parameters for the Hausman test when the estimators are from different sets of equations

Hausman (1978) developed a widely-used model specification test that has passed the test of time. In this paper, we show that the asymptotic variance of the difference of the two estimators can be a singular matrix. Three illustrative examples are used, namely an exogeneity test for the linear regression model, a test for the Box–Cox transformation, and a test for sample selection bias.     

Estimating VARs with long stationary heterogeneous panels: A comparison of the fixed effect and the mean group estimators

The most promising candidates for estimating vector autoregressive models with long, stationary, possibly heterogeneous panel data sets (panel-VARs) are the fixed effect (FE) and the mean group (MG) estimators. With a view to providing guidance to applied researchers on how to pool in a panel-VAR, this paper compares the performance of the FE and the MG estimators both asymptotically and in Monte Carlo simulations. The main results of the analysis suggest the use of both estimators in applied work. If FE and MG estimates give similar estimates, the FE estimator should be used because it is more efficient. If they differ, the MG estimator should be used only if the panel is sufficiently long — say, twice as long as usually recommended in the dynamic panel data literature. If FE and MG estimates differ and the panel is not long enough, neither is generally a desirable alternative and other estimation techniques are needed.