Following up on this post, estimating the consumption function.
Consider the canonical consumption-income relationship discussed in macro textbooks. For pedagogical reasons, the relationship is often stated as:
(1) C = c0 + c1 Yd
Where C is real consumption and Yd is real disposable income. Figure 1 depicts the relationship over the 1967-2015Q2 period.
Figure 1: Consumption (blue) and disposable personal income (red), in billions of Ch.2009$, SAAR. NBER defined recession dates shaded gray. Source: BEA, 2015Q2 advance release, and NBER.
Now consider the corresponding figure, in logs.
Figure 2: Log consumption (blue) and log disposable personal income (red), in billions of Ch.2009$, SAAR. NBER defined recession dates shaded gray. Source: BEA, 2015Q2 advance release, and NBER.
It does seem hard to choose one over the other merely by looking. Reader Mike V writes:
I just think you lose a lot of people by using logs for every. graph.
Which one is the better way to characterize the relationship? At first glance, estimating each by way of OLS does not allow much to distinguish between the two. In levels:
(2) C = -336.7 + 0.945Yd
R2 = 0.999, SER = 93.26, Nobs = 194, DW = 0.56. Bold Face indicates significance at 5% msl using HAC standard errors.
In logs:
(3) c = -0.651 + 1.061yd
R2 = 0.999, SER = 0.014, Nobs = 194, DW = 0.48. Bold Face indicates significance at 5% msl using HAC standard errors.
Clearly, neither specification is adequate, but is one to be preferred to another? Theory does not provide guidance, as the linear consumption function is typically used for convenience.
One factor one can use to inform a choice is heteroscedasticity, the characteristic wherein the variance of the errors is not constant. One does not observe the true residuals, but one can examine the squared estimated residuals, and see if there is a systematic pattern between the squared residuals and the right hand side variable. Figure 3 presents the squared residuals from the levels specification, and Figure 4 presents squared residuals from the log specification.
