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Stata Technical Bulletin

33


In the rest of this insert, we briefly describe each type of coefficient listed by Iistcoef. Full details along with citations to
original sources are found in Long (1997). For purposes of illustration, we use an example with only two independent variables.

Standardized coefficients

It is often useful to compute coefficients after some or all of the variables have been standardized to a unit variance. This
is particularly useful for the models where the scale of the dependent variable is arbitrary (e.g., logit, probit). By default,
estimation commands express coefficients in the metric of the variables as they are found in the dataset. Standardization can be
introduced as follows.

The linear regression model estimated by regress can be expressed as

У = βo + βιxι + β2X2 + ɛ                                    (1)

The independent variables can be standardized with simple algebra. Let σ>s be the standard deviation of x>s. Then, dividing each
.17,. by <7fe and multiplying the corresponding ∕¾. by
σj. gives

У = βo + (σ'∙'ι)--h (σ^2∕¾)--h ε

ɑi         σ%

= σfe∕⅞ is an x-standardized coefficient. For a continuous variable, <iβ' can be interpreted as follows. For a standard
deviation increase in ж&,
у is expected to change by <iβ' units, holding all other variables constant.

To standardize for the dependent variable, let σy be the standard deviation of y. We can standardize у by dividing (1) by
<Γj,, thus giving

У βo l βι l ∕¾ l ɛ

σy σy σy σy σy

Then βky =βk∕<Xy is a y-standardized coefficient that can be interpreted as follows. For a unit increase in ж&, у is expected to
g

change by ββy standard deviations, holding all other variables constant.

For a dummy variable, the interpretation would be as follows. Having characteristic x⅛ (as opposed to not having the
g
characteristic) results in an expected change in y of βkv standard deviations, holding all other variables constant.

It is also possible to standardize both y and the afs as in

У_ _ ∕⅜ + (ɑɪffiλ a:ɪ + (σ2β2 ж2 + ɪ

σy   σy σy / σι σy ) σ2 σy

Then, βk =(σkβk)∕<Xy is a fully standardized coefficient that can be interpreted as follows. For a standard deviation increase in
Xj., y is expected to change by βk standard deviations, holding all other variables constant.

A variety of other models (logit, tobit, cnreg, intreg, probit, ologit, oprobit) can be written as

У* = βo + βιxι + β2X2 + ε                                    (2)

where y* is a latent variable.

In models with a latent dependent variable, (2) can be divided by σy*. To estimate the variance of the latent variable, the
quadratic form is used.

Var(y*) = ∕3'Var(it)∕3 + Var(e)

where β is a vector of estimated coefficients and Var(ж) is the covariance matrix for the ж’8 computed from the observed data.
In some models such as tobit, Var(e) is estimated from the data. In probit models, Var(e) = 1 by assumption and Var(e) = τr2∕3
in logit models.

Factor and percent change

In logit-based models and models for counts, coefficients can be expressed either as a factor or multiplicative change in the
odds or the expected count, or as a percentage change. In logit the or option computes the factor change in the odds, referred
to as odds ratios. In mlogit, the rrr option computes the relative risk ratio which is also a factor change in the odds.



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