= Econometrics Notation =



== Observations and Measurements ==

The number of observations is ''n''.

The outcome variable is ''y''. The outcome measurement for observation ''i'' is ''y,,i,,''.

If there is a single predictor, it may be specified as ''x''; the measurement is ''x,,i,,''. More commonly, there is a set of predictors specified like ''x,,1,,'', ''x,,2,,'', and so on. The measurements are then ''x,,1i,,'', ''x,,2i,,'', and so on.

When expressing data with [[LinearAlgebra|linear algebra]], the outcome measurements are composed into vector ''y'' with size ''n'', and the predictor measurements are composed into matrix '''''X''''' of shape ''n'' by ''p''.

A very common exception: income is usually represented by ''Y'' or ''y''. In relevant literature, expect to see different letters.



== Error Terms ==

Error terms are variably represented by ''ε'', ''e'', ''u'', or ''v''. The error term for observation ''i'' would be represented like ''ε,,i,,''.



== Statistics ==

There is a mixture of notations for scalar statistics. The conventional estimators for population mean ''μ'', variance ''σ^2^'', standard deviation ''σ'', covariance ''σ,,xy,,'', and correlation ''ρ,,xy,,'' are:

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{{attachment:variance.svg}}

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Frequently for multiple variable statistics, some pieces of [[LinearAlgebra|linear algebra]] notation are introduced. For example, covariances are frequently expressed in a covariance matrix. Covariances of ''x'' and ''y'' are specified as ''σ,,xy,,''; variances are expressed as covariances of ''x'' and ''x''.

{{attachment:covariancem.svg}}



== Distributions ==

The [[Statistics/NormalDistribution|normal distribution]] is frequently expressed in econometrics. The typical notation is ''x,,i,, ~ N(μ, σ)''.

For multiple variables, at minimum the distribution is specified as ''NI'' to emphasize independence of the distributions. Some pieces of [[LinearAlgebra|linear algebra]] notation are also introduced. For example, the joint statement of [[Econometrics/Exogeneity|exogeneity]] and [[Econometrics/Homoskedasticity|homoskedasticity]] is:

{{attachment:exo.svg}}

Note how the covariance matrix is fully expressed as the [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] of each term's variance.



== Modeling ==

A univariate model is specified with a constant term ''α'' and a coefficient term ''β''. A multivariate model of ''j'' variables specifies constant ''β,,0,,'' and coefficients ''β,,1,,'' through ''β,,j,,''. A [[LinearAlgebra|linear algebra]] notation uses a coefficient vector ''β'' of size ''p''.

In any case, when a model is estimated, the estimated coefficients are notated differently. Scalar notations attach a hat, as in ''βˆ,,0,,''. The linear algebra notation replaces ''β'' with ''b''.

The predicted outcome from a model is also marked as an estimate by attaching a hat: ''yˆ''.

The generic calculation of the residual for observation ''i'' is ''y,,i,, - yˆ,,i,,''. The sum of square residuals (SSR) is what is minimized to fit a model.

And the coefficient of determination is:

{{attachment:rsquared.svg}}



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