Log Income vs. Linear Income: An Application of the Encompassing Principle*

Abstract

An open question in empirical economics is whether models should be estimated by using the actual, or linear, values of economic variables or their logarithms. This paper applies the principle of encompassing to suggest specification and misspecification tests of log vs. linear individual equations fitted to I (1) data, and illustrates the analysis for US quarterly disposable income. The finite-sample properties of the encompassing tests are examined in a Monte Carlo experiment customized to the parameter values found in the empirical analysis.

The effect of mortgage interest deduction and mortgage characteristics on house prices

“…We are indeed unable to reject the null hypothesis of no cointegration in some of the countryspecific cointegration tests between the logarithmic transformed variables (results not reported here). The logarithmic transformation is often applied to stabilize the variance of a time series (Ltkepohl and Xu, 2012) or to linearize an exponential growing trend (Ermini and Hendry, 2008). A log transformation of our house price series in differences would, however, increase the variance of the series (especially in the 1980s).…”

Commodity Prices and Unit Root Tests

“…Consequently, deflating by an upward trending price index introduces a downward trend in real prices. Ermini and Hendry (1996) applied the encompassing principle to U.S. quarterly disposable income and illustrated the consequences of the choice between log income and observed income. Corradi and Swanson (2006) also discuss the impact of the choice of the level versus log-level observations.…”

Simulation Encompassing: Testing Non‐nested Hypotheses*

“…But an explicit expression for δ α 0 often does not exist in practice. The simple example of linear vs. log‐linear models discussed here and in Ermini and Hendry (2008) shows that even when the explicit expression for δ α 0 exists it may involve a disproportionate amount of calculation. However, these difficulties can be avoided using a simulation algorithm based on the expansions in to estimate as is now shown.…”

“…As a specific illustration of the methods described above consider again a case which has been widely investigated in the literature, i.e. the testing of linear vs. log‐linear models (see Ermini and Hendry, 2008). The Cox test statistic for this case has been investigated by Aneuyrn‐Evans and Deaton (1980), who derive its form assuming symmetric truncation of the normal distribution of ε t .…”

“…The method is illustrated by the example of comparing a linear model with a log‐linear model. This example, although appearing to be simple, involves all the typical computational difficulties for non‐nested hypothesis testing (see Ermini and Hendry, 2008, for further analysis of this example). The simulation approach discussed in this paper provides an improved approximation to the non‐nested test statistics with better small sample properties, in that their distributions are closer to their asymptotic distributions.…”