In statistics, the Ramsey Regression Equation Specification Error Test (RESET) test is a general specification test for the linear regression model. More specifically, it tests whether non-linear combinations of the explanatory variables help to explain the response variable. The intuition behind the test is that if non-linear combinations of the explanatory variables have any power in explaining the response variable, the model is misspecified in the sense that the data generating process might be better approximated by a polynomial or another non-linear functional form.
The test was developed by James B. Ramsey as part of his Ph.D. thesis at the University of Wisconsin–Madison in 1968, and later published in the Journal of the Royal Statistical Society in 1969. [1] [2]
Consider the model
The Ramsey test then tests whether has any power in explaining y. This is executed by estimating the following linear regression
and then testing, by a means of an F-test whether through are zero. If the null-hypothesis that all coefficients are zero is rejected, then the model suffers from misspecification.
In statistics, the Gauss–Markov theorem states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be normal for the theorem to apply, nor do they need to be independent and identically distributed.
In statistics, the logistic model is a statistical model that models the log-odds of an event as a linear combination of one or more independent variables. In regression analysis, logistic regression is estimating the parameters of a logistic model. Formally, in binary logistic regression there is a single binary dependent variable, coded by an indicator variable, where the two values are labeled "0" and "1", while the independent variables can each be a binary variable or a continuous variable. The corresponding probability of the value labeled "1" can vary between 0 and 1, hence the labeling; the function that converts log-odds to probability is the logistic function, hence the name. The unit of measurement for the log-odds scale is called a logit, from logistic unit, hence the alternative names. See § Background and § Definition for formal mathematics, and § Example for a worked example.
Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables. This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. Take the typical supply and demand model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand and then set the price.
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.
In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term (endogenous), in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable and is not correlated with the error term, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the input dataset and the output of the (linear) function of the independent variable. Some sources consider OLS to be linear regression.
In statistics, the Vuong closeness test is a likelihood-ratio-based test for model selection using the Kullback–Leibler information criterion. This statistic makes probabilistic statements about two models. They can be nested, strictly non-nested or partially non-nested. The statistic tests the null hypothesis that the two models are equally close to the true data generating process, against the alternative that one model is closer. It cannot make any decision whether the "closer" model is the true model.
Cointegration is a statistical property of a collection (X1, X2, ..., Xk) of time series variables. First, all of the series must be integrated of order d (see Order of integration). Next, if a linear combination of this collection is integrated of order less than d, then the collection is said to be co-integrated. Formally, if (X,Y,Z) are each integrated of order d, and there exist coefficients a,b,c such that aX + bY + cZ is integrated of order less than d, then X, Y, and Z are cointegrated. Cointegration has become an important property in contemporary time series analysis. Time series often have trends—either deterministic or stochastic. In an influential paper, Charles Nelson and Charles Plosser (1982) provided statistical evidence that many US macroeconomic time series (like GNP, wages, employment, etc.) have stochastic trends.
In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term. The distinction between endogenous and exogenous variables originated in simultaneous equations models, where one separates variables whose values are determined by the model from variables which are predetermined. Ignoring simultaneity in the estimation leads to biased estimates as it violates the exogeneity assumption of the Gauss–Markov theorem. The problem of endogeneity is often ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations. Instrumental variable techniques are commonly used to mitigate this problem.
The Chow test, proposed by econometrician Gregory Chow in 1960, is a test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test for the presence of a structural break at a period which can be assumed to be known a priori. In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.
In statistics, the Breusch–Pagan test, developed in 1979 by Trevor Breusch and Adrian Pagan, is used to test for heteroskedasticity in a linear regression model. It was independently suggested with some extension by R. Dennis Cook and Sanford Weisberg in 1983. Derived from the Lagrange multiplier test principle, it tests whether the variance of the errors from a regression is dependent on the values of the independent variables. In that case, heteroskedasticity is present.
In statistics, semiparametric regression includes regression models that combine parametric and nonparametric models. They are often used in situations where the fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. Semiparametric regression models are a particular type of semiparametric modelling and, since semiparametric models contain a parametric component, they rely on parametric assumptions and may be misspecified and inconsistent, just like a fully parametric model.
In statistics, model specification is part of the process of building a statistical model: specification consists of selecting an appropriate functional form for the model and choosing which variables to include. For example, given personal income together with years of schooling and on-the-job experience , we might specify a functional relationship as follows:
Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median of the response variable. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met.
An error correction model (ECM) belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long-run common stochastic trend, also known as cointegration. ECMs are a theoretically-driven approach useful for estimating both short-term and long-term effects of one time series on another. The term error-correction relates to the fact that last-period's deviation from a long-run equilibrium, the error, influences its short-run dynamics. Thus ECMs directly estimate the speed at which a dependent variable returns to equilibrium after a change in other variables.
In statistics, errors-in-variables models or measurement error models are regression models that account for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured exactly, or observed without error; as such, those models account only for errors in the dependent variables, or responses.
In econometrics, the Park test is a test for heteroscedasticity. The test is based on the method proposed by Rolla Edward Park for estimating linear regression parameters in the presence of heteroscedastic error terms.
In statistics, linear regression is a statistical model which estimates the linear relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models.
In least squares estimation problems, sometimes one or more regressors specified in the model are not observable. One way to circumvent this issue is to estimate or generate regressors from observable data. This generated regressor method is also applicable to unobserved instrumental variables. Under some regularity conditions, consistency and asymptotic normality of least squares estimator is preserved, but asymptotic variance has a different form in general.
