econometrics-en-blog

[econometrics-en-blog]

Breusch–Pagan test

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. 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. Suppose that we estimate the regression model y = β 0 + β 1 x + u , {\displaystyle y=\beta _{0}+\beta _{1}x+u,\,} and obtain from this fitted model a set of values for u ^ {\displaystyle {\hat {u}}} , the residuals. Ordinary least squares constrains these so that their mean is 0 and so, given the assumption that their variance does not depend on the independent variables, an estimate of this variance can be obtained from the average of the squared values of the residuals. If the assumption is not held to be true, a simple model might be that the variance is linearly related to independent variables. Such a model can be examined by regressing the squared residuals on the independent variables, using an auxiliary regression equation of the form u ^ 2 = γ 0 + γ 1 x + v . {\displaystyle {\hat {u}}^{2}=\gamma _{0}+\gamma _{1}x+v.\,} This is the basis of the Breusch–Pagan test. If an F-test confirms that the independent variables are jointly significant then the null hypothesis of homoskedasticity can be rejected. The Breusch–Pagan test tests for conditional heteroskedasticity.[citation needed] It is a chi-squared test:[clarification needed] the test statistic is nχ2 with k degrees of freedom. It tests the null hypothesis of homoskedasticity. If the Chi Squared value is significant with p-value below an appropriate threshold (e.g. p<0.05) then the null hypothesis of homoskedasticity is rejected and heteroskedasticity assumed. If the Breusch–Pagan test shows that there is conditional heteroskedasticity, the original regression can be corrected by using the Hansen method[citation needed], using robust standard errors, or re-thinking the regression equation by changing and/or transforming independent variables. More details Android, Windows

[econometrics-en-blog]

Breusch–Godfrey test

In statistics, the Breusch–Godfrey test, named after Trevor S. Breusch and Leslie G. Godfrey, is used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series. In particular, it tests for the presence of serial correlation that has not been included in a proposed model structure and which, if present, would mean that incorrect conclusions would be drawn from other tests, or that sub-optimal estimates of model parameters are obtained if it is not taken into account. The regression models to which the test can be applied include cases where lagged values of the dependent variables are used as independent variables in the model's representation for later observations. This type of structure is common in econometric models. Because the test is based on the idea of Lagrange multiplier testing, it is sometimes referred to as LM test for serial correlation. A similar assessment can be also carried out with the Durbin–Watson test and the Ljung–Box test. More details Android, Windows

[econometrics-en-blog]

Stationary process

In mathematics and statistics, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance, if they are present, also do not change over time. Since Stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data is often transformed to become stationary. The most common cause of violation of Stationarity are trends in mean, which can be due either to the presence of a unit root or of a deterministic trend. In the latter case the process is called trend stationary process, stochastic shocks have only transitory effects, and the process is mean-reverting (on a mean which changes deterministically over time). On the contrary, in the first case stochastic shocks have permanent effects and the process is not mean-reverting. A trend stationary process is not strictly stationary, but can easily be made such by removing the underlying trend (function solely of time). Similarly, processes with one or more unit roots can be made stationary through differencing. An important type of non-stationary process that does not include a trend-like behavior is the cyclostationary process. A "stationary process" is not the same thing as a "process with a stationary distribution".[clarification needed] Indeed, there are further possibilities for confusion with the use of "stationary" in the context of stochastic processes; for example a "time-homogeneous" Markov chain is sometimes said to have "stationary transition probabilities". Besides, all stationary Markov random processes are time-homogeneous. More details Android, Windows

[econometrics-en-blog]

Heteroscedasticity-consistent standard errors

The topic of heteroscedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression as well as time series analysis. These are also known as Eicker–Huber–White standard errors (also Huber–White standard errors or White standard errors), to recognize the contributions of Friedhelm Eicker, Peter J. Huber, and Halbert White. In regression and time-series modelling, basic forms of models make use of the assumption that the errors or disturbances ui have the same variance across all observation points. When this is not the case, the errors are said to be heteroscedastic, or to have heteroscedasticity, and this behaviour will be reflected in the residuals u i ^ {\displaystyle \scriptstyle {\widehat {u_{i}}}} estimated from a fitted model. Heteroscedasticity-consistent standard errors are used to allow the fitting of a model that does contain heteroscedastic residuals. The first such approach was proposed by Huber (1967), and further improved procedures have been produced since for cross-sectional data, time-series data and GARCH estimation. More details Android, Windows

[econometrics-en-blog]

Newey–West estimator

A Newey–West estimator is used in statistics and econometrics to provide an estimate of the covariance matrix of the parameters of a regression-type model when this model is applied in situations where the standard assumptions of regression analysis do not apply. It was devised by Whitney K. Newey and Kenneth D. West in 1987, although there are a number of later variants. The estimator is used to try to overcome autocorrelation (also called serial correlation), and heteroskedasticity in the error terms in the models, often for regressions applied to time series data. The problem in autocorrelation, often found in time series data, is that the error terms are correlated over time. This can be demonstrated in Q ∗ {\displaystyle Q*} , a matrix of sums of squares and cross products that involves σ ( i j ) {\displaystyle \sigma _{(ij)}} and the rows of X {\displaystyle X} . The least squares estimator b {\displaystyle b} is a consistent estimator of β {\displaystyle \beta } . This implies that the least squares residuals e i {\displaystyle e_{i}} are "point-wise" consistent estimators of their population counterparts E i {\displaystyle E_{i}} . The general approach, then, will be to use X {\displaystyle X} and e {\displaystyle e} to devise an estimator of Q ∗ {\displaystyle Q*} . This means that as the time between error terms increases, the correlation between the error terms decreases. The estimator thus can be used to improve the ordinary least squares (OLS) regression when the residuals are heteroskedastic and/or autocorrelated. w ℓ = 1 − ℓ L + 1 {\displaystyle w_{\ell }=1-{\frac {\ell }{L+1}}} ^ "Newey West estimator – Quantitative Finance Collector".  ^ Newey, Whitney K; West, Kenneth D (1987). "A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix". Econometrica. 55 (3): 703–708. doi:10.2307/1913610. JSTOR 1913610.  ^ Andrews, Donald W. K. (1991). "Heteroskedasticity and autocorrelation consistent covariance matrix estimation". Econometrica. 59 (3): 817–858. doi:10.2307/2938229. JSTOR 2938229.  ^ Newey, Whitney K.; West, Kenneth D. (1994). "Automatic lag selection in covariance matrix estimation". Review of Economic Studies. 61 (4): 631–654. doi:10.2307/2297912. JSTOR 2297912.  ^ Smith, Richard J. (2005). "Automatic positive semidefinite HAC covariance matrix and GMM estimation". Econometric Theory. 21 (1): 158–170. doi:10.1017/S0266466605050103.  ^ Greene, William H. (1997). Econometric Analysis (3rd ed.).  More details Android, Windows

[econometrics-en-blog]

Simultaneous equations model

Simultaneous equation models are a type of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics. More details Android, Windows

[econometrics-en-blog]

Distributed lag

In statistics and econometrics, a distributed lag model is a model for time series data in which a regression equation is used to predict current values of a dependent variable based on both the current values of an explanatory variable and the lagged (past period) values of this explanatory variable. The starting point for a distributed lag model is an assumed structure of the form y t = a + w 0 x t + w 1 x t − 1 + w 2 x t − 2 + . . . + error term {\displaystyle y_{t}=a+w_{0}x_{t}+w_{1}x_{t-1}+w_{2}x_{t-2}+...+{\text{error term}}} or the form y t = a + w 0 x t + w 1 x t − 1 + w 2 x t − 2 + . . . + w n x t − n + error term , {\displaystyle y_{t}=a+w_{0}x_{t}+w_{1}x_{t-1}+w_{2}x_{t-2}+...+w_{n}x_{t-n}+{\text{error term}},} where yt is the value at time period t of the dependent variable y, a is the intercept term to be estimated, and wi is called the lag weight (also to be estimated) placed on the value i periods previously of the explanatory variable x. In the first equation, the dependent variable is assumed to be affected by values of the independent variable arbitrarily far in the past, so the number of lag weights is infinite and the model is called an infinite distributed lag model. In the alternative, second, equation, there are only a finite number of lag weights, indicating an assumption that there is a maximum lag beyond which values of the independent variable do not affect the dependent variable; a model based on this assumption is called a finite distributed lag model. In an infinite distributed lag model, an infinite number of lag weights need to be estimated; clearly this can be done only if some structure is assumed for the relation between the various lag weights, with the entire infinitude of them expressible in terms of a finite number of assumed underlying parameters. In a finite distributed lag model, the parameters could be directly estimated by ordinary least squares (assuming the number of data points sufficiently exceeds the number of lag weights); nevertheless, such estimation may give very imprecise results due to extreme multicollinearity among the various lagged values of the independent variable, so again it may be necessary to assume some structure for the relation between the various lag weights. The concept of distributed lag models easily generalizes to the context of more than one right-side explanatory variable. More details Android, Windows

[econometrics-en-blog]

Probit model

In statistics, a probit model is a type of regression where the dependent variable can only take two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, if estimated probabilities greater than 1/2 are treated as classifying an observation into a predicted category, the probit model is a type of binary classification model. A probit model is a popular specification for an ordinal or a binary response model. As such it treats the same set of problems as does logistic regression using similar techniques. The probit model, which employs a probit link function, is most often estimated using the standard maximum likelihood procedure, such an estimation being called a probit regression. Probit models were introduced by Chester Bliss in 1934; a fast method for computing maximum likelihood estimates for them was proposed by Ronald Fisher as an appendix to Bliss' work in 1935. More details Android, Windows

[econometrics-en-blog]

Generalized least squares

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model. GLS can be used to perform linear regression when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1934. More details Android, Windows

[econometrics-en-blog]

Generalized method of moments

In econometrics, the generalized method of moments (GMM) is a generic method for estimating parameters in statistical models. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data may not be known, and therefore maximum likelihood estimation is not applicable. The method requires that a certain number of moment conditions were specified for the model. These moment conditions are functions of the model parameters and the data, such that their expectation is zero at the true values of the parameters. The GMM method then minimizes a certain norm of the sample averages of the moment conditions. The GMM estimators are known to be consistent, asymptotically normal, and efficient in the class of all estimators that do not use any extra information aside from that contained in the moment conditions. GMM was developed by Lars Peter Hansen in 1982 as a generalization of the method of moments which was introduced by Karl Pearson in 1894. Hansen shared the 2013 Nobel Prize in Economics in part for this work. More details Android, Windows

[econometrics-en-blog]

Multicollinearity

In statistics, multicollinearity (also collinearity) is a phenomenon in which two or more predictor variables in a multiple regression model are highly correlated, meaning that one can be linearly predicted from the others with a substantial degree of accuracy. In this situation the coefficient estimates of the multiple regression may change erratically in response to small changes in the model or the data. Multicollinearity does not reduce the predictive power or reliability of the model as a whole, at least within the sample data set; it only affects calculations regarding individual predictors. That is, a multiple regression model with correlated predictors can indicate how well the entire bundle of predictors predicts the outcome variable, but it may not give valid results about any individual predictor, or about which predictors are redundant with respect to others. In case of perfect multicollinearity the design matrix is singular and therefore cannot be inverted. Under these circumstances, for a general linear model y = X β + ϵ {\displaystyle y=X\beta +\epsilon } , the ordinary least-squares estimator β ^ O L S = ( X ⊤ X ) − 1 X ⊤ y {\displaystyle {\hat {\beta }}_{OLS}=(X^{\top }X)^{-1}X^{\top }y} does not exist. Note that in statements of the assumptions underlying regression analyses such as ordinary least squares, the phrase "no multicollinearity" is sometimes used to mean the absence of perfect multicollinearity, which is an exact (non-stochastic) linear relation among the regressors. More details Android, Windows

[econometrics-en-blog]

Ordered logit

In statistics, the ordered logit model (also ordered logistic regression or proportional odds model), is a regression model for ordinal dependent variables, first considered by Peter McCullagh. For example, if one question on a survey is to be answered by a choice among "poor", "fair", "good", "very good", and "excellent", and the purpose of the analysis is to see how well that response can be predicted by the responses to other questions, some of which may be quantitative, then ordered logistic regression may be used. It can be thought of as an extension of the logistic regression model that applies to dichotomous dependent variables, allowing for more than two (ordered) response categories. More details Android, Windows

[econometrics-en-blog]

Moving-average model

See also: Moving average In time series analysis, the moving-average (MA) model is a common approach for modeling univariate time series. The moving-average model specifies that the output variable depends linearly on the current and various past values of a stochastic (imperfectly predictable) term. Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure. The moving-average model should not be confused with the moving average, a distinct concept despite some similarities. Contrary to the AR model, the MA model is always stationary. More details Android, Windows

[econometrics-en-blog]

Multiple correlation

In statistics, the coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions that can be computed linearly from the predictive variables. The coefficient of multiple correlation takes values between 0 and 1; a higher value indicates a better predictability of the dependent variable from the independent variables, with a value of 1 indicating that the predictions are exactly correct and a value of 0 indicating that no linear combination of the independent variables is a better predictor than is the fixed mean of the dependent variable. The coefficient of multiple correlation is computed as the square root of the coefficient of determination, but under the particular assumptions that an intercept is included and that the best possible linear predictors are used, whereas the coefficient of determination is defined for more general cases, including those of nonlinear prediction and those in which the predicted values have not been derived from a model-fitting procedure. More details Android, Windows

[econometrics-en-blog]

Least squares

The result of fitting a set of data points with a quadratic function Conic fitting a set of points using least-squares approximation The method of least squares is a standard approach in regression analysis to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation. The most important application is in data fitting. The best fit in the least-squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the independent variable (the x variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and non-linear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. The non-linear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases. Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve. When the observations come from an exponential family and mild conditions are satisfied, least-squares estimates and maximum-likelihood estimates are identical. The method of least squares can also be derived as a method of moments estimator. The following discussion is mostly presented in terms of linear functions but the use of least-squares is valid and practical for more general families of functions. Also, by iteratively applying local quadratic approximation to the likelihood (through the Fisher information), the least-squares method may be used to fit a generalized linear model. For the topic of approximating a function by a sum of others using an objective function based on squared distances, see least squares (function approximation). The least-squares method is usually credited to Carl Friedrich Gauss (1795), but it was first published by Adrien-Marie Legendre. More details Android, Windows

[econometrics-en-blog]

Method of moments

See method of moments (probability theory) for an account of a technique for proving convergence in distribution. In statistics, the method of moments is a method of estimation of population parameters. One starts with deriving equations that relate the population moments (i.e., the expected values of powers of the random variable under consideration) to the parameters of interest. Then a sample is drawn and the population moments are estimated from the sample. The equations are then solved for the parameters of interest, using the sample moments in place of the (unknown) population moments. This results in estimates of those parameters. The method of moments was introduced by Karl Pearson in 1894. More details Android, Windows

[econometrics-en-blog]

Maximum likelihood estimation

This article is about the statistical techniques. For computer data storage, see Partial response maximum likelihood. In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model given observations, by finding the parameter values that maximize the likelihood of making the observations given the parameters. MLE can be seen as a special case of the maximum a posteriori estimation (MAP) that assumes a uniform prior distribution of the parameters, or as a variant of the MAP that ignores the prior and which therefore is unregularized.[citation needed] The method of maximum likelihood corresponds to many well-known estimation methods in statistics. For example, one may be interested in the heights of adult female penguins, but be unable to measure the height of every single penguin in a population due to cost or time constraints. Assuming that the heights are normally distributed with some unknown mean and variance, the mean and variance can be estimated with MLE while only knowing the heights of some sample of the overall population. MLE would accomplish this by taking the mean and variance as parameters and finding particular parametric values that make the observed results the most probable given the model. In general, for a fixed set of data and underlying statistical model, the method of maximum likelihood selects the set of values of the model parameters that maximizes the likelihood function. Intuitively, this maximizes the "agreement" of the selected model with the observed data, and for discrete random variables it indeed maximizes the probability of the observed data under the resulting distribution. Maximum likelihood estimation gives a unified approach to estimation, which is well-defined in the case of the normal distribution and many other problems. More details Android, Windows