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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 x₁, x₂, and so on. The measurements are then x_1i, x_2i, 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 ε_i.

Statistics

Distributions

The 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 linear algebra notation are also introduced. For example, the joint statement of exogeneity and homoskedasticity is:

Note how the covariance matrix is fully expressed as the 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 σ², standard deviation σ, covariance σ_xy, and correlation ρ_xy are:

Based on OLS regression, the estimated outcome for observation i is:

No matter the regression method, the residual is:

And the coefficient of determination, a.k.a. the R², is:

CategoryRicottone

Statistics/EconometricsNotation (last edited 2025-05-17 03:45:12 by DominicRicottone)

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  Comment: Regression
+   ← Revision 10 as of 2024-06-07 15:14:21 → ⇥
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-== Data ==
+== Observations and Measurements ==
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-The outcome variable is ''y''. For observation ''i'', the outcome value is ''y,,i,,''.
+The outcome variable is ''y''. The outcome measurement for observation ''i'' is ''y,,i,,''.
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-The treatment variable is ''x,,1,,''. For observation ''i'', the treatment value is ''x,,1i,,''.
+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.
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-The control variables are ''x,,2,,'' through ''x,,k,,'' (up to ''k'' - 1 control variables). For observation ''i'', a control value might be ''x,,2i,,''.
+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,,''.
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-The average outcome is:
+== 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:
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-The variance is:
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-The standard deviation is:
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-The covariance between the treatment and outcome is:
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-The correlation between the treatment and outcome is:
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+Based on [[Econometrics/OrdinaryLeastSquares|OLS regression]], the estimated outcome for observation ''i'' is:
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+{{attachment:estimate.svg}}
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-== Regression ==
+No matter the regression method, the residual is:
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-A regression line passes through two points:
+{{attachment:residual.svg}}
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-{{attachment:regression1.svg}}
+And the coefficient of determination, a.k.a. the ''R^2^'', is:
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-and

{{attachment:regression2.svg}}

Take the generic equation form of a line:

{{attachment:b01.svg}}

Insert the first point into this form.

{{attachment:b02.svg}}

This can be trivially rewritten to solve for ''a'' in terms of ''b'':

{{attachment:b03.svg}}

Insert the second point into the original form.

{{attachment:b04.svg}}

Now additionally insert the solution for ''a'' in terms of ''b''.

{{attachment:b05.svg}}

Expand all terms to produce:

{{attachment:b06.svg}}

This can now be eliminated into:

{{attachment:b07.svg}}

Giving a solution for ''b'':

{{attachment:b08.svg}}

This solution is trivially rewritten as:

{{attachment:b09.svg}}

Expand the formula for correlation as:

{{attachment:b10.svg}}

This can now be eliminated into:

{{attachment:b11.svg}}

Finally, ''b'' can be eloquently written as:

{{attachment:b12.svg}}

Giving a generic formula for the regression line:

{{attachment:b13.svg}}
+{{attachment:rsquared.svg}}

Diff for "Statistics/EconometricsNotation"

Econometrics Notation

Observations and Measurements

Error Terms

Statistics

Distributions

Statistics