Ratios of Parameters: Some Econometric Examples

• Author: Jenny Lye,Joe Hirschberg
• Published date: 01 December 2018
• Date: 01 December 2018
• DOI: http://doi.org/10.1111/1467-8462.12300

For the Student

Ratios of Parameters: Some Econometric Examples

Jenny Lye and Joe Hirschberg*

Abstract

Ratios of parameter estimates are often used in

econometric applications. However, construct-

ing conf‌idence intervals (CIs) for these ratios

can cause diff‌iculties since the ratio of

asymptotically normally distributed random

variables are Cauchy distributed and thus have

no f‌inite moments. This article presents a

method for the estimation of CIs based on the

Fieller approach that has been shown to be

preferable to the usual Delta method. Using

example applications in Stata and R, we

demonstrate that a few extra steps in the

examination of the estimate of the ratio can

provide a CI with superior coverage.

1. Introduction

Many econometric applications draw inferences

from a parameter of interest that is def‌ined as

the ratio of regression coeff‌icients. However,

the properties of the ratio of estimates can be

problematic. This article examines eight examples

of econometric models where inferences for ratios

are used. These examples include: the interpreta-

tion of a dummy variable in terms of changes in

a continuous regressor; the location of a turning

point in a quadratic specif‌ication where the

marginal impact of a regressor changes sign;

the interpretation of the marginal effect of one

regressor when interacted with another; the

estimation of the long-run effect in a dynamic

model; the determination of the ‘Taylor Rule’for

dynamic macro models; the determination of the

non-accelerating inf‌lation rate of unemployment

(NAIRU); the willingness to pay; and the

structural parameter in an exactly identif‌ied

system of equations as estimated by the two-stage

least squares method. We present analysis of

applications of each of these cases along with the

corresponding Stata code to obtain these results.

In this article, we emphasise the use of

conf‌idence intervals (CIs) instead of the use of

p-values. Because CIs are expressed in the units

of the parameter of interest instead of as a

probabilistic abstraction, we concentrate on the

nature of these intervals. The traditional approach

for constructing CIs is based on the Delta method

where a f‌irst-order Taylor-series expansion is used

to approximate a linear relationship between the

estimated parameters and the ratio. This method is

the standard approach for the estimation of CIs

and tests of hypothesis for non-linear functions of

regression parameter estimates and is available in

most econometric/statistics software.

* Lye and Hirschberg: Department of Economics,

University of Melbourne, Victoria 3010 Australia. Corre-

sponding author: Lye, email <jnlye@unimelb.edu.au>.

The Australian Economic Review, vol. 51, no. 4, pp. 578–602 DOI: 10.1111/1467-8462.12300

°

C2018 The University of Melbourne, Melbourne Institute: Applied Economic & Social Research,

Faculty of Business and Economics

Published by John Wiley & Sons Australia, Ltd

An alternative method for the construction of

CIs for ratios is Fieller’s (1954) proposal. This

approach has been shown to be superior to

the application of the Delta method in several

applications (e.g., Hirschberg and Lye 2010c).

This superiority has been found in the coverage of

the resulting tests where the estimated 100ð1

aÞ%interval from the Fieller method more

closely coincides with the theoretical interval

than the alternate Delta interval.

1

In addition,

Fieller intervals are not forced to be symmetric as

are the Delta intervals. However, in some cases,

the Fieller approach may notresult in a f‌inite CI for

some values of a. In such cases the resulting CI

may also be the complement of a f‌inite interval or

the whole real line. The advantage of the graphical

approach presented here is that it can provide an

indication of when this may be the case and can

provide partial information that can be used to

construct open-ended intervals. This is shown in

some of the examples below.

Another attraction of the Fieller method is

the ease of computation. We demonstrate that

this method can be computed using a direct

computational method similar to the Delta and

does not require the use of simulations and

sampling strategies as would be needed when

employing a Bootstrap or Bayesian method.

In the next section, we outline the aspects of the

theory involved for the Fieller as it can be applied

to the ratios of regression coeff‌icients and discuss

the syntax of Stata commands we employ to

construct the CIs. In Section 3 we provide eight

examples of econometric applications where the

result of interest is a ratio of parameters or the ratio

of linear combinations of parameters as estimated

from standard regressions, probit models, systems

of equations and non-linear models.

2. Conf‌idence Intervals for a Ratio of

Regression Coeff‌icients

2.1 Ratios of Regression Parameters

Consider the general linear model

Y¼XBðÞþeð1Þ

where Yis an T1ðÞvector of observations

on the dependent variable, XBðÞis a T1ðÞ

function of an TkðÞmatrix Xof regressors

and a k1ðÞvector Bof unknown parameters,

and eis an T1ðÞvector of disturbances with

mean 0 and TTðÞcovariance matrix V.

Suppose interest lies in the ratio of regression

coeff‌icients def‌ined by

u¼r

f¼H0Bþh

L0Bþl;ð2Þ

where Hand Lare k1ðÞvectors of known

constants, thus H0Band L0Bare linear combina-

tions of the parameters and kand lare constants.

The usual estimate of the ratio uin equation (2) is

^

u¼^r

^

f¼H0B

^þh

L0^

Bþl

;ð3Þ

where ^

Bis the k1ðÞvector of the estimates

of Bin equation (1).

The models we consider here assume that

the estimates of Bare normally distributed with

a covariance matrix that we can approximate.

In general, regression, maximum likelihood

estimation and non-linear least squares all

result in estimates of the parameters under

these assumptions.

2

2.2 The Delta Method for CIs

The estimated variance of ^

ubased on the Delta

method is def‌ined for the ratio of the parameter

estimates. We assume that the parameter

estimates are bivariate normally distributed:

3

^r

^

f

"#

Nr

f;

s2

1s12

s12 s2

2

"#()

ð4Þ

and ^

u¼^r=^

f.

The estimate of the variance of the ratio of

normally distributed random variables based

on the Delta method is def‌ined as (see example

5.5.27 in Casella and Berger 2002)

varð^

uÞ ^r=^

f



2s2

1=^r2



þs2

2=^

f2

h

2s12=^

f^r



 ^

f2s2

1þ^

u2s2

22^

us12

hi

ð5Þ

Lye and Hirschberg: Ratios of Parameters 579

°

C2018 The University of Melbourne, Melbourne Institute: Applied Economic & Social Research, Faculty of Business and

Economics