Consumption, Real Interest Rates, and Habit Formation

Many macroeconomists use models of aggregate consumption based on utility maximization by a rational representative agent. Some models also include liquidity constrained or rule of thumb consumers, but all micro founded models include inter-temporal substitution subject to a lifetime budget constraint. It is also standard to assume that utility is additively separable in consumption and everything else. In the simplest models, the utility function is time separable with a constant inter-temporal elasticity of substitution. Finally it is standard to assume that economic variables have stationary distributions around a balanced growth path. All this implies that the expected rate of growth of consumption is a constant (the inter temporal elasticity of substitution of consumption) times the expected real interest rate. The inter-temporal elasticity of substitution can easily be estimated with a regression of the rate of growth of consumption on the expectable real interest rate, that is with an instrumental variables regression of the rate of growth of consumption on the achieved real interest rate in which lagged variables are used as instruments.

Estimates of this constant are alarmingly tiny (generally around 0.1). The estimates imply extremely slow growth of consumption given measured real interest rates and even complete patience (zero subjective discounting). That is estimates based on first differences are inconsistent with estimates based on long term trends.

In order to reconcile the model with the data, it is necessary to modify the assumptions in a way which changes the short term correlation without changing the long term trend (or vice versa). This was an important impetus for the development of models of rational addiction, that is, of habit formation. Here the argument is that the marginal utility of consumption depends on the relationship between current consumption and past consumption. One way to interpret the models is that all consumption is addictive so doing without something to which one has become accustomed is very painful. Another is that people have to learn how to consume so high consumption becomes more efficient. A neutral description is that consumption changes slowly because of habit formation.

The standard new Keynesian DSGE model due to Smets and Wouters (based on the Eichenbaum, Christiano and Evans model) includes habit formation, as required to explain the low correlation of the rate of growth of consumption and the expectable real interest rate. This means that the short term evidence is consistent with the existence of a long term trend of increasing consumption. As an aside, it also means that the short term evidence of low correlation between the rate of growth of consumption and the real interest rate, is consistent with the belief that the key to long run growth is a high after tax real interest rate. I note in passing that this belief is not solidly based on evidence, but is held with great confidence by many economists.

Their claim must be that short run fluctuations in real interest rates don’t matter much, because of habit formation, but long run persistent variation matters a lot. This claim suggests a simple empirical investigation. Their argument must be that the average rate of growth of consumption over long periods of time is highly correlated with the average real interest rate over those periods. So as consumption growth and real interest rates are averaged over longer and longer periods of time, their correlation gets higher until the very strong long run association appears. This is an implication of models of habit formation.

As far as I know, no one has checked. In fact, it the regression coefficient of the change in log consumption on the average real interest rate does not increase markedly as the interval of time increases. This is true whether achieved real interest rates or real interest rates predicted using lagged variables are used. It is also true whether or not interest is compounded.

A few regressions

The regression of the change in seasonally quarterly log personal consumption expenditures (dlrcons) on the achieved real interest rate (ri3) — the real interest rate is the 3 month t-bill rate corrected for the change in the personal consumption expenditures deflator.

. reg dlrcons ri3

Number of obs = 267
R-squared = 0.0061

dlrcons Coef. t

ri3 .0242031 (1.27)
_cons .0079801 (14.59)

The coefficient on the real interest rate is tiny. It is extraordinarily small partly because this is a regression of consumption growth on the achieved real interest rate rather than a predicted real interest rate. To predict the real interest rate I use the fitted values (pri) from a regression of the real interest rate on the real interest rate lagged two quarters, the real interest rate lagged three quarters and the nominal interest rate lagged two quarters

. reg dlrcons pri3

Number of obs = 264
R-squared = 0.0082

dlrcons Coef. t
pri3 .0437429 (1.47)
_cons .0077503 (12.57)

Now consider changes over a year so, for example, one observation of the dependent variable (sdlrc4) is the change in log consumption is from the second quarter of 2000 to the second quarter of 2001 and the explanatory variable (sri4) is the sum of the real interest rates paid from the third quarter of 2000 through the second quarter of 2001. Since overlapping one year intervals are used, the standard errors are Newey West autocorrelation corrected standard errors with three lags.

Regression with Newey-West standard errors Number of obs = 264
maximum lag: 3

sdlrc4 Coef. t

rin4 .0427988 (1.87)
_cons .0311175 (13.18)

The coefficient estimated with overlapping annual changes is slightly higher than the coefficient estimated with quarterly changes. This is a bit of evidence of habit formation. However, the coefficient is still tiny. This pattern would correspond to extremely slowly changing habits such that a year is still a brief interval.

Now a regression with overlapping two year periods

Regression with Newey-West standard errors Number of obs = 260
maximum lag: 7

sdlrc8 Coef. t

rin8 .0470761 (2.02)
_cons .0619985 (13.57)

The coefficient estimated with two year long intervals is almost identical to the coefficient estimated with one year long intervals. There is no further evidence of habit formation. The now medium term correlation of real interest rates and consumption growth remains miniscule.

Now consider overlapping five year intervals

Regression with Newey-West standard errors Number of obs = 248
maximum lag: 19

sdlrc20 Coef. t

rin20 .0511966 (2.04)
_cons .1540758 (11.33)

Again extending the interval has almost no effect. There is essentially no sign of a high long term correlation between real interest rates and consumption growth, that is no sign that the low quarterly correlation is due to habit formation.

Importantly interest paid over 5 years has enough variance to identify the coefficient. Simulations of standard macroeconomic models would probably not show a large and precisely estimated coefficient, because fluctutations of real interest rates are not persistent in such models. They certainly wouldn’t yield a tiny and precisely estimated coefficient. In fact there have been huge and highl persistent fluctuations in US achieved safe short term real interest rates with enormous rates in the 80s and very high rates in the 90s. The fact that the growth rate of aggregate consumption was similar in the 80s to that of other decades should have made it obvious that standard macroeconomic models with habit formation did not fit the data at all.