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== 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. == 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: |
There is a mixture of notations for scalar statistics. The conventional estimators for population mean ''μ'', [[Statistics/Variance|variance]] ''σ^2^'', standard deviation ''σ'', [[Statistics/Covariance|covariance]] ''σ,,xy,,'', and [[Statistics/Correlation|correlation]] ''ρ,,xy,,'' are: |
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| Based on [[Econometrics/OrdinaryLeastSquares|OLS regression]], the estimated outcome for observation ''i'' is: | 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''. |
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| {{attachment:estimate.svg}} | {{attachment:covariancem.svg}} |
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| No matter the regression method, the residual is: | |
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| {{attachment:residual.svg}} | |
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| And the coefficient of determination, a.k.a. the ''R^2^'', is: | == Distributions == The [[Statistics/NormalDistribution|normal distribution]] is commonly used in econometrics, and a shorthand notation has emerged as ''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. == Models == A linear model of ''k'' variables specifies constant ''β,,0,,'' and coefficients ''β,,1,,'' through ''β,,k,,''. The [[LinearAlgebra|linear algebra]] notation uses a coefficient vector ''β'' of size ''p''. When the model is fit using [[Statistics/OrdinaryLeastSquares|regression]], the estimated coefficients are notated using a hat, as in ''ˆβ,,0,,''. The linear algebra notation uses ''b'' instead. The predicted outcome is also notated using a hat, as in ''ŷ,,i,,''. The generic calculation of the [[Statistics/Residuals|residual]] for observation ''i'' is ''y,,i,, - ŷ,,i,,''. The residual sum of squares (RSS) is what is minimized to fit a model. And the coefficient of determination is: |
Econometrics Notation
Observations and Measurements
The number of observations is n.
The outcome variable is y. The outcome measurement for observation i is yi.
If there is a single predictor, it may be specified as x; the measurement is xi. More commonly, there is a set of predictors specified like x1, x2, and so on. The measurements are then x1i, x2i, and so on.
When expressing data with 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 εi.
Statistics
There is a mixture of notations for scalar statistics. The conventional estimators for population mean μ, variance σ2, standard deviation σ, covariance σxy, and correlation ρxy are:
Frequently for multiple variable statistics, some pieces of linear algebra notation are introduced. For example, covariances are frequently expressed in a covariance matrix. Covariances of x and y are specified as σxy; variances are expressed as covariances of x and x.
Distributions
The normal distribution is commonly used in econometrics, and a shorthand notation has emerged as xi ~ N(μ, σ).
For multiple variables, at minimum the distribution is specified as NI to emphasize independence of the distributions. Some pieces of linear algebra notation are also introduced.
Models
A linear model of k variables specifies constant β0 and coefficients β1 through βk. The linear algebra notation uses a coefficient vector β of size p.
When the model is fit using regression, the estimated coefficients are notated using a hat, as in ˆβ0. The linear algebra notation uses b instead.
The predicted outcome is also notated using a hat, as in ŷi.
The generic calculation of the residual for observation i is yi - ŷi. The residual sum of squares (RSS) is what is minimized to fit a model.
And the coefficient of determination is:
