This paper is based on the material in Romer's Sections 7.1 and 7.2.
The literature review that opens the paper is not as relevant for you as for Hall's readers in 1978 because you aren't familiar with the papers he cites and are not reading the paper with the same pre-concieved ideas about what empirical consumption functions should look like. The prevailing empirical models used real consumption spending as the dependent variable and current and lagged values of disposable income as independent explanatory variables. On page 972, Hall points out that this is problematic for two reasons: (1) current disposable income is "endogenous," meaning that shocks to aggregate consumption almost surely affect aggregate disposable income, which means that the explanatory variable (income) is endogenous and injects bias into the estimates of the coefficients in a standard regression, and (2) the pattern of lagged effects of income on consumption is almost surely unstable, depending on the nature of the income change and how people perceive it (for example, temporary vs. permanent changes).
The central claim on which Hall's test is based is that modern consumption theory predicts that consumption will be a "random walk." A random walk is a variable whose change from period to period (apart from normal trend growth) is random and cannot be predicted based on the past values of the variable. In the simplest terms, people formed a lifetime consumption plan at time t - 1 based on all the information available at t - 1 about their future income (and everything else). This planned lifetime consumption path might have an upward trend so that consumption at t would be predictably above that at t - 1. (The slope of the consumption path depends on the interest rate and the rate of time preference, as in Romer's equation (2.21).) If households receive no new information about their lifetime budget constraint at time t that wasn't known at t - 1, then they will just proceed up their planned lifetime consumption path in period t, and the change in consumption will just be the trend amount. If they get new information at time t, then they will adjust their lifetime consumption path going forward and consumption in t will differ from the trend path planned in t - 1. Thus, Hall argues that the only things that should affect consumption in t are consumption in t - 1 plus trend change (reflecting the path planned in t - 1) and new information arriving in period t. Everything known at t - 1 should already be embodied in the t - 1 plan and thus should not have any separate affect on consumption in t beyond what is already incorporated in consumption at t - 1. This is the basis of the tests described at the beginning of Section II.
Hall's theoretical model is very similar to Romer's Chapter 2 models. He uses finite rather than infinite lifetimes and annual rather than continuous compounding of interest/utility (as in the Diamond model, but with longer lifetimes). Hall's delta (δ) corresponds to Romer's rho (ρ) as the rate of time preference. When he uses the constant elasticity of substitution utility function, his sigma (σ) parameter plays the role of Romer's theta (θ) . Hall's sigma is equivalent to the reciprocal of Romer's theta.
The latter part of Section II can be largely ignored. Focus on the results in Sections III through VI.
Questions for analysis
If there are no random shocks, how does Hall's Corollary 2 (p. 974) relate to Romer's equations (2.47) and (2.21)?
Hall's estimates of gamma (γ) in Table 1 are very close to one. Do you expect this? Why?
Why would including expenditures on consumer durables be problematic for Hall's tests?
Summarize the results of Hall's tests in Sections III through VI. What conclusions does he draw about the validity of the modern consumption model? Do you think that these conclusions are supported by his evidence?