Learning Econometrics Using GAUSS
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Last updated: 02 Jan As we're restricting to unbiased estimators, minimum mean squared error implies minimum variance. We calculate:. We calculate. The generalized least squares GLS , developed by Aitken ,  extends the Gauss—Markov theorem to the case where the error vector has a non-scalar covariance matrix.
This assumption is considered inappropriate for a predominantly nonexperimental science like econometrics. The dependent variable is assumed to be a linear function of the variables specified in the model.
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The specification must be linear in its parameters. This does not mean that there must be a linear relationship between the independent and dependent variables. The independent variables can take non-linear forms as long as the parameters are linear. Data transformations are often used to convert an equation into a linear form.
For example, the Cobb—Douglas function —often used in economics—is nonlinear:.
Ordinary least squares
But it can be expressed in linear form by taking the natural logarithm of both sides: . This assumption also covers specification issues: assuming that the proper functional form has been selected and there are no omitted variables. One should be aware, however, that the parameters that minimize the residuals of the transformed equation not necessarily minimize the residuals of the original equation. This assumption is violated if the explanatory variables are stochastic, for instance when they are measured with error , or are endogenous.
Instrumental variable techniques are commonly used to address this problem. A violation of this assumption is perfect multicollinearity , i. One scenario in which this will occur is called "dummy variable trap," when a base dummy variable is not omitted resulting in perfect correlation between the dummy variables and the constant term. Multicollinearity as long as it is not "perfect" can be present resulting in a less efficient, but still unbiased estimate. The estimates will be less precise and highly sensitive to particular sets of data.
ISBN 13: 9780471510741
The outer product of the error vector must be spherical. This implies the error term has uniform variance homoscedasticity and no serial dependence.
Heteroskedasticity occurs when the amount of error is correlated with an independent variable. For example, in a regression on food expenditure and income, the error is correlated with income. Low income people generally spend a similar amount on food, while high income people may spend a very large amount or as little as low income people spend.
Heteroskedastic can also be caused by changes in measurement practices.
For example, as statistical offices improve their data, measurement error decreases, so the error term declines over time. This assumption is violated when there is autocorrelation. Autocorrelation can be visualized on a data plot when a given observation is more likely to lie above a fitted line if adjacent observations also lie above the fitted regression line. Autocorrelation is common in time series data where a data series may experience "inertia.
The first two assumptions are self-explanatory; the parameters we are estimating must be linear, and our sample data is to be collected through a randomized, probabilistic mechanism. The third condition, no perfect collinearity, ensures that the regressors are not perfectly correlated with one another.
An example of this is including both outcomes of a binary variable into a model. Suppose we are interested in official language preferences: if we were to add English and French as regressors, the model would exhibit perfect collinearity because we know if someone prefers English, they do not prefer French at the exact same time. Mathematically, if they were both indicator variables, we would not be able to differentiate when an observation prefers English or French because one of them will always have a value of 1.
Exogeneity means that the regressors cannot be correlated with the error term.click here
Gauss Programming For Econometricians And Financial Analysts
The converse of this is endogeneity, and examples of this include omitted variable bias, reverse causality, and measurement error. The fifth and final assumption is homoscedasticity, which means the variance of the error term must be constant no matter what the value of regressors are. However, as the first article alluded to a few weeks ago, we need to use econometric models with discretion. To put the importance of these assumptions into perspective, consider this analogy.
The criminal code is in place so that the citizens of our country can function well together without harming one another. A police officer will never come up to you and ask you to recite the criminal code, but when you start violating the laws, you will likely find yourself in trouble. It is important for us to identify when we are breaking the law, and find methods to avoid doing so.
The same can be said using OLS. By learning the five assumptions, we know of possible issues that we may run into when performing linear regression. In later articles, we will discuss specific ways to mitigate violations of these conditions. For example, when we have endogeneity present the fourth assumption is violated , our OLS estimator will be biased.