0 1990 American Statistical Association Journal of Business & Economic Statistics, July 1990, Vol. 8, No. 3

Permanent Income, Current Income, and Consumption

John Y. Campbell Woodrow Wilson School, Princeton University, Princeton, NJ 08544

N. Gregory Mankiw Department of Economics, Harvard University, Cambridge, MA 02138

This article reexamines the consistency of the permanent-income hypothesis with aggregate postwar U.S. data. The permanent-income hypothesis is nested within a more general model in which a fraction of income accrues to individuals who consume their current income rather than their permanent income. This fraction is estimated to be about 50%, indicating a substantial departure from the permanent-income hypothesis. Our results cannot be easily explained by time aggregation or small-sample bias, by changes in the real interest rate, or by nonsepara- bilities in the utility function of consumers.

KEY WORDS: Euler equation; Instrumental variables; Monte Carlo study; Nonseparable utility; Real interest rate; Time aggregation.

1. INTRODUCTION crues to individuals who consume their current income, During the past decade, much effort has been di- while the remainder (1 - /.) accrues to individuals who rected at the question of whether the response of con- consume their permanent income. [Similar ideas were sumption to income is consistent with the permanent- explored by DeLong and Summers (1986), Flavin income hypothesis. Hall (1978) showed that a central (1981), Hall and Mishkin (1982), Hayashi (1982), and implication of the theory is that consumption should Summers (1982).] We argue that the model is consistent follow a random walk. He argued that, to a first ap- with the empirical findings in the existing literature. We proximation, postwar U.S. data are consistent with this show how to estimate A and test the permanent-income implication. In contrast, Flavin (1981) reported that hypothesis that i = 0 using an instrumental variables consumption is "excessively sensitive" to income, a con- (IV) approach. Our test is valid whether or not income clusion that has been widely interpreted as evidence has a unit root, and it is more powerful than the stan- that liquidity constraints are important for understand- dard unrestricted test for consumption following a ran- ing consumer spending (Dornbusch and Fischer 1987). dom walk. By lagging our instruments two periods, we Yet Mankiw and Shapiro (1985) showed that Flavin's are able to avoid econometric difficulties that would procedure for testing the permanent-income hypothesis otherwise be created by time aggregation of our data. can be severely biased toward rejection if income has We also show how to test our framework against an approximately a unit root. Nelson (1987) recently reap- even more general time series representation for con- praised the evidence on the permanent-income hypoth- sumption and income. esis and concluded that it is generally favorable. The second goal of this article is to generalize the Other recent research has examined the permanent- preceding approach to handle alternative versions of income theory from a different point of view. Campbell the permanent-income hypothesis. We can allow for (1987) studied the implications of the theory for savings changes in the real interest rate (Bean 1986; Grossman behavior, and Campbell and Deaton (1989) and West and Shiller 1981; Hall 1988; Hansen and Singleton 1983; (1988), following Deaton (1987), looked at its impli- Mankiw 1981). We can also allow for nonseparability cations for the "smoothness" (the standard deviation in the utility function between consumption and other of the change) of consumption. These works argued goods. Following previous work, we examine interac- that. although some of the qualitative implications of tions with labor supply (Bean 1986; Eichenbaum, Han- the model are fulfilled, consumption appears to be too sen, and Singleton 1988; Mankiw, Rotemberg, and smooth and there is weak evidence that saving moves Summers 1985), durable goods (Bernanke 1985; Startz too little to be consistent with the theory. 1986), and government purchases (Aschauer 1985; Bai- The first goal of this article is to provide a simple ley 1971; Bean 1986; Kormendi 1983). We examine framework for understanding these disparate results. whether these alternative formulations of preferences We nest the permanent-income hypothesis in a more can explain the apparent excess sensitivity of consump- general model in which some fraction of income i.ac- tion to income. 266 Journal of Business & Economic Statistics, July 1990

The organization of the article is as follows. Section unpredictable, then there are no instruments that are 2 describes our model and IV test procedure in more orthogonal to E, but correlated with AY,, and the pro- detail and relates our approach to the existing literature. cedure breaks down. In this case, permanent income Section 3 reports empirical results for the basic model. and current income are equal, so the parameter 3. is Section 4 presents some Monte Carlo results to shed unidentified. More generally, if AY, is only slightly pre- light on the finite-sample properties of our tests. Section dictable, it will be hard to obtain a precise estimate of 5 extends the model to allow for the effects of time- the parameter 3.. varying real interest rates and nonseparabilities in the Equation (I), estimated by IV, can be thought of as utility function. Section 6 concludes, and a brief Ap- a restricted version of a more general two-equation sys- pendix gives some details about the construction of our tem in which AC, and AY, are regressed directly on the Monte Carlo experiment. instruments. If we have K instruments, Z,, through ZKt, then the general system is 2. AN INSTRUMENTAL VARIABLES APPROACH TO THE PERMANENT-INCOME HYPOTHESIS Consider an economy in which there are two groups of agents, who receive disposable income Y,, and Y2, Total disposable income Y, is just the sum of the dis- The permanent-income hypothesis implies that the vec- posable income of these two groups; Y, = Y1, + Y2,. tor P = 0 (i.e., PI = ... = PK= 0). This can be tested We assume that the first group receives a fixed share i directly, and without any need for predictability of AY,, of total disposable income, so Y,, = iY, and Y2, = by OLS estimation of the first equation of (2). But as (1 - I*)Y,. Flavin (1981) argued, it is hard to interpret a rejection Agents in the first group consume their current dis- of the permanent-income hypothesis in this framework; posable income, so c,,= Y,,. Taking first differences, an estimate of i,is much more informative about the AC,, = AY,, = I.AY,. Agents in the second group, by economic importance of deviations from the theory. For contrast, consume their permanent disposable income; this reason, we focus on IV estimation of (1). C,, = Y;, = (1 - /Z)Y;P.By the argument of Hall (1978), When there is more than a single instrument, Equa- as elaborated by Flavin (1981), we then have LC2, = tion (1) places overidentifying restrictions on (2). These p + (1 - i)~,,where ,LL is a constant and E, is the state that predictable changes in consumption and in- - innovation between time t 1 and time t in agents' come, and therefore the vectors P and ;), are propor- assessment of total permanent income Yf. Since E, is an tional to one another (p = 3.y, or Pl!y, = ... = BK!~lK innovation, it is orthogonal to any variable that is in = 3.). We regard the presence of overidentifying re- agents' information set at time t - 1. strictions as an advantage of our framework over that The change in aggregate consumption can now be of Flavin (1981), which includes enough extra lags to written as make the model just-identified. But if we are to put much weight on the estimate of I. that we obtain from AC, = AC,, + AC,, = p + 1-AY, + (1 - ).)el. (1) (I), it is important to test the restrictions. A simple way Our empirical strategy will be to estimate i and test the to do this is to add K - 1 instruments as right-side permanent-income hypothesis that i = 0 by running variables in the IV regression and to test the joint sig- the regression (1). Note, however, that (1) cannot be nificance of these extra variables. We shall use this Wald estimated by ordinary least squares (OLS). The error test. term E, is orthogonal to lagged variables but not nec- An alternative is a Lagrange multiplier (LM) test. essarily to AY,, so an OLS estimate of A will generally One can regress the residual from the IV regression on be inconsistent. the instruments and then compare T times the R2from It is tempting to argue that AY, and E, are positively this regression, where T is the sample size, with the X' correlated so that the OLS estimate is upward biased distribution with (K - 1) df. We use the R' from the and gives an upper bound on the true value of 3.. This residual regression as an informal diagnostic statistic, is not necessarily true for general multivariate time se- but we do not use the LM approach to test the model ries processes for Y,, however. In the Appendix, we because it is harder to generalize this approach to han- present a stylized model of consumption and income in dle conditional heteroscedasticity and autocorrelation which AY, and E, can be negatively correlated, causing in the equation error. downward bias in the OLS estimate of I.. Equation (1) also implies that for any value of I. the The solution to this problem is to estimate (1) by IV R2 of the regression of AC, on instruments must be less rather than OLS. Any lagged stationary variables are than the R2 of the regression of AY, on instruments, potentially valid instruments, since they are orthogonal unless E, and AY, are strongly negatively correlated. [To to el if the model is correct. Of course, good instruments see this, note that the R2 in the consumption equation must also be correlated with AY,. If AY, is completely is Ir2var(Z,y)/(A2var(AYr)+ (1 - 3.)2var(~,)+ 2I.(l - Campbell and Mankiw: Permanent Income, Current Income, and Consumption 267 i.)cov(AY,, e,)), which is less than or equal to the R2 in income hypothesis so that it can explain why saving the income equation when (1 - i.)?var(e,) + 2A(1 - seems to be insufficiently variable, as found by Camp- i.)cov(AY,, 8,) 2 0.1 This means that a small R2 for bell (1987). Finally, our model can explain the other- changes in consumption cannot be interpreted as strong wise puzzling smoothness of consumption growth evidence in favor of the permanent-income hypothesis. (Campbell and Deaton 1989; Deaton 1987; Flavin 1988; If the R2for changes in income is small, it is very pos- West 1988). More details on this point are given in sible that consumption is close to a random walk as Section 4. measured by R', but the permanent-income hypothesis is far from true as measured by the coefficient i. 3. ISSUES OF SPECIFICATION AND The choice of instruments is critically important in EMPIRICAL RESULTS FOR our approach. Perhaps the most obvious instruments THE BASIC MODEL are those that summarize the history of Y,. Flavin (1981) used lagged values of detrended Y, in her test of the Before we can estimate our model, we need to ad- model. Mankiw and Shapiro (1985), however, showed dress several issues of specification that arise from the that this leads to statistical problems when the Y, process nature of the aggregate time series on consumption and has a unit root. Lagged values of AY, are valid instru- income. ments but, as we show, they do not explain a large We have not yet distinguished between different com- fraction of the variance of AY, because the univariate ponents of consumption. In fact. the permanent-income time series process for Y, is close to a random walk. hypothesis is normally tested using consumption ex- Campbell (1987) emphasized that the history of C, penditure only on nondurables and services, excluding should also provide good instruments for AY,. This is expenditure on durables. The implicit assumption is because, according to the permanent-income hypoth- that permanent-income consumers have a utility func- esis. C, summarizes agents' information about the future tion that is separable between durable and nondurable of the Y, process. If agents have better information goods so that consumption of nondurables and services about Y, than is contained in that variable's own history. can be treated without also modeling purchases of du- then C, will help to forecast Y,. Lagged levels of C, rable goods. cannot be used as instruments because they are non- We begin by making this same assumption so that we stationary: the permanent-income hypothesis and our can use nondurables and services consumption in our alternative model imply. however, that C, and Y, are tests. (In Sec. 5, we bring durables back into the anal- cointegrated (Engle and Granger 1987) so that savings. ysis.) In our approach, however, the use of a component S, = Y, - C,, is stationary. Lagged values of S, or AC, of consumption causes two problems. Unless we rescale are likely to increase the precision with which the pa- the data. the coefficient I. will be the fraction of income rameter i. can be estimated. accruing to current-income consumers times the share When lagged AY,, AC,, and S, are used as instru- in consumption of nondurables and services, and the ments. the unrestricted system (2) becomes an error- difference between the consumption measure and in- correction model for consumption and income of the come will be nonstationary. We adopt a simple solution, type proposed by Davidson, Hendry, Srba, and Yeo which is to multiply nondurables and services con- (1978) and Davidson and Hendry (1981). They inter- sumption by the mean ratio of total consumption to preted their error-correction models in terms of dis- nondurables and services consumption (1.12 in our equilibrium adjustment of consumption to income; our data). This multiplication does not affect any of the approach suggests an alternative interpretation, involv- statistical tests (except those using lagged S as an in- ing forward-looking consumption behavior of at least strument), but it preserves our original interpretation some agents. As discussed previously, our interpreta- of A. tion places testable restrictions on the error-correction Our discussion so far has been couched in terms of framework. levels and differences of the raw series C, and Y,. This Financial variables are also appealing instruments. is appropriate if these series follow homoscedastic linear There is considerable evidence that changes in stock processes in levels, with or without unit roots. In fact, prices and interest rates help to forecast changes in however, aggregate time series on consumption and income (Fischer and Merton 1984; Litterman and Weiss income appear to be closer to log-linear than linear. 1985; Sims 1980). Hall (1978) found that stock prices The mean change and the innovation variance both also forecast changes in consumption. We use both grow with the level of the series. A correction of some stock prices and interest rates in our empirical work. sort appears necessary. Our model (1) has the potential to reconcile several One approach. which was used by Campbell and Dea- of the empirical results in the existing literature. It dis- ton (1989), is to divide AC, and AY, by the lagged level plays Flavin's (1981) "excess sensitivity" of consump- of income, Y,.,. This scaling method is used in Table 2, tion to income, since consumption moves more closely Section 3.1. with income when i. > 0. Our model also makes saving A second approach is to take logs of all the variables equal to (1 - i.) times its value under the permanent- in the previous section. (Hereafter, we use lowercase 268 Journal of Business & Economic Statistics, July 1990 letters to denote log variables.) Equation (1) should Third, there may be white-noise errors in the levels of hold in logs when 2 = 0 if aggregate consumption is our consumption and income variables. These could be chosen by a representative agent with a power-utility due to "transitory consumption" or to measurement function facing a constant riskless real interest rate errors. White-noise errors in levels become first-order (Bean 1986; Hall 1988; Hansen and Singleton 1983; moving average errors in our differenced specification Nelson 1987). The instruments discussed in Section 2 and could be correlated with once-lagged instruments, remain stationary, but we now use the difference be- but they cannot be correlated with twice-lagged instru- tween log consumption and log income, the log con- ments. sumption-income ratio, rather than saving. These arguments for twice-lagging our instruments This approach has the advantage that the log equation also imply that the error terms in Equations (1)and (2) generalizes straightforwardly to cases in which the real have a first-order moving average structure. If we ig- interest rate varies or the utility function is nonsepar- nore this and use standard OLS and IV procedures, the able, whereas the levels equation does not. The diffi- coefficient estimates remain consistent but the standard culty with taking logs is that the parameter i. can no errors are inconsistent. Fortunately, a straightforward longer be precisely interpreted as the fraction of agents standard-error correction is available (White 1984); who consume their current income; if one is willing to White's methods can also be used to allow for condi- approximate the log of an average by an average of tional heteroscedasticity in the error terms of (1) and logs, however, the interpretation of the model is not (2). We report heteroscedasticity-consistent and au- substantially affected. The log scaling method is used tocorrelation-consistent standard errors, but for our in Table 3, Section 3.2. data these corrections make almost no difference. Tak- Another data problem is that consumption and in- ing account of the moving average error structure tends come are measured as quarterly averages rather than to reduce the reported standard errors very slightly; at points in time. If the permanent-income hypothesis taking account of heteroscedasticity tends to increase holds in continuous time, then measured consumption them very slightly. is the time average of a random walk. The results of Working (1960) imply that it will have a first-order serial 3.1 Basic Empirical Results correlation of .25, which could lead us to reject the model even if it is true. Christiano, Eichenbaum, and To estimate our model, we use standard U.S. quar- Marshall (1987) and Hall (1988) argued that a contin- terly time series data, obtained from the Data Re- uous-time version of the permanent-income model fits sources, Inc., data bank. In the model, Y, is measured postwar U.S. data better than a discrete-time version. as disposable income per capita in 1982 dollars, and C, We deal with this problem by lagging the instruments is per capita consumption of nondurables and services more than one period so that there is at least a two- in 1982 dollars. Our data set runs from 1948: 1 through period time gap between the instruments and the vari- 1985:4. We begin our sample in 1953: 1, the date used ables in Equation (1). The time average of a continuous- by Blinder and Deaton (1985), Campbell (1987), and time random walk is uncorrelated with all variables Campbell and Deaton (1989), which avoids the Korean lagged more than one period, so by using twice-lagged War. The data are reproduced in Table 1. instruments we obtain a test of the model that is valid Table 2 reports results for the 1953 : 1-1985 :4 sample for time-averaged data. period. The table has six columns. The first gives the The extra lag in the instruments also helps meet sev- row number and the second the instruments used. (A eral other potential objections. First, Goodfriend constant term is always included as both an instrument (1986) noted that aggregate variables are not in indi- and a regressor but is not reported in the tables.) The viduals' information sets contemporaneously because of third and fourth columns give the adjusted R2statistics delays in government publication of aggregate statistics. for OLS regressions of LC,/ Y,., and hY,/ Y,.,, respec- Since such delays are typically no more than a few tively, on the instruments. In parentheses we report the months, lagging the instruments an extra quarter largely p value for a Wald test of the hypothesis that all coef- avoids this problem. (The problem is not completely ficients in these regressions are 0 except the intercept. avoided, since the data are revised over a long period The fifth column gives the IV estimate of i,with an of time. We use as instruments financial variables such asymptotic standard error. The final column gives the as nominal interest rates, which are known contem- adjusted R2 statistic for an OLS regression of the re- poraneously, and this fully circumvents the problem.) sidual from the IV regression on the instruments. In Second, it is sometimes suggested that those goods parentheses we report the p value for a Wald test of labeled nondurable in the National Income Accounts the overidentifying restrictions placed by Equation (1) are, in fact, partly durable. Durability would introduce on the general system (2). a first-order moving average term into the change in The first row of Table 2 shows that we obtain a coef- consumer expenditure (Mankiw 1982); this would not ficient of about .3 when we estimate Equation (1) by affect our procedure using twice-lagged instruments. OLS. The remaining rows give IV results for various Campbell and Mankiw: Permanent Income. Current Income, and Consumption 269

Table 1. Data Used in Model Estimation

Income Consumption Interest Labor Durables Government Date 0') (c) Deflator (i) supply stock spending 270 Journal of Business & Economic Statistics, July 1990

Table 1. Continued

Income Consumption Interest Labor Durables Government Date iY) iC) Deflator ii) supply stock spending Campbell and Mankiw: Permanent Income, Current Income, and Consumption 271

Table 1. Continued

Income Consumption Interest Labor Durables Government Date 0') (c) Deflator (i) supply stock spending

NOTE lncome IS total d~sposableIncome per caplta In 1982 dollars Consumpt~onIS consumptlon of nondurables and sewlces per caplta In 1982 dollars The deflator IS the ratlo of nomlnal consumptlon of nondurables and sewlces per caplta to the consumptlon measure In 1982 dollars The Interest rate IS a three-month treasury-b~llrate Labor supply IS man-hours per caplta In nonagr~culaturalestablishments The stock of durable goods IS computed In two steps We flrst constructed an end-of-quarter per caplta stock serles from the annual stock at the beg~nn~ng of the sample perlod and the serles on consumer durable purchases In 1982 dollars assumlng a deprec~at~onrate of 6% per quarter We then measured the stock of consumer durables dur~nga quarter as the average of the end-of-quarter stock and the prevlous end-of-quarter stock Government spend~ngIS measured as total government purchases per caplta In 1982 dollars choices of instruments. In all cases we include at least the permanent-income hypothesis. The permanent-in- lags 2-4 of the instruments; in some rows we add lags come hypothesis is rejected particularly strongly when 5 and 6 for a total of five instruments. lags 2-6 of consumption changes or nominal interest- The main conclusions we draw from Table 2 are as rate changes are used as instruments. The adjusted R2 follows. First, each set of instruments has some fore- statistics for consumption are quite small in absolute casting power for scaled income changes two quarters terms, but they often exceed the adjusted R2 for in- ahead, but this forecasting power is particularly modest come. when income changes themselves are used as instru- Previously, we argued that our model (1) would nor- ments (the adjusted R2statistics are only in the range mally imply a smaller R2for consumption growth than 2% to 4%), reflecting the fact that the univariate in- for income growth. The predictability of consumption come process is close to a random walk. Better fore- growth is therefore surprisingly high. One possible ex- casting power is obtained from consumption changes, planation is that income growth is measured with error. interest rates, and saving; together these variables de- Measurement error uncorrelated with our instruments liver an adjusted R2statistic of more than 11% in row will not bias the IV estimate of 2 but will reduce the 9. We also tried using lagged changes in real stock prices forecastability of income growth. as instruments but found no forecasting power for either Third, our IV procedure estimates the parameter i. consumption growth or income growth. Hall's (1978) to be between .3 and .7. The t statistic on i.is more finding that stock prices forecast consumption appears than 2.8 in every row of the table except the second. to be due to his inclusion of the first lagged stock price. The significance level for the IV test of the permanent- Second, several instrument sets have significant fore- income hypothesis always exceeds the significance level casting power for consumption; this is evidence against for the corresponding OLS test. 272 Journal of Business & Economic Statistics, July 1990

Table 2. Basic Permanent-Income Model (scaled levels)

OLS regressions on Z Instruments i estimate Test of Row (z) AC equation AY equation (standard error) restrictions

None (OLS)

NOTE Th~stable reports ~nstrumental-var~ablesestimates of Equat~on(1) tn the text 1C = ii + $AY, uslng tnstrurnental vartables Z The var~ablesC Y and S have been d~v~dedby the flrst lag of Y The statlsttcs In columns 3 and 4 are adjusted R2 values from OLS regressions of AC and AY on Z [Eq (2)] and s~gn~flcancelevels for tests of the hypothes~sthat all coefflctents except the constant are 0 (In parentheses) The statlsttcs tn column 5 are the instrumental var~ables estlmate of , wtth an asymptottc standard error (In parentheses) Column 6 glves the adjusted R' from a regresslon of the IV res~dualonto the Instruments and the slgnlflcance level for a Wald test of the over~dent~fyngrestrlctlons of the IV model All standard errors and test statlstlcs are heteroscedasttc~tyand autocor- relatlon consistent Finally, there is no evidence against our model (1) rest of the data. [The special NSLI payments were ana- with a free parameter i. The tests in the last column lyzed in several early works on the permanent-income of Table 2 uniformly fail to reject the model against the model, with inconclusive results. See Mayer (1972) for more general alternative (2).~, a review.) When he exclude 1950: 1 from the sample, we get 3.2 How Robust Are the Results? similar results to those reported for 1953-1985. The The results of Table 2 are robust to the measures of only comparable episode in the 1953-1985 period is the consumption and income used. We obtain extremely temporary tax rebate of 1975 :2, which caused dispos- similar results when we use consumption and income able income to grow 4.6%, or 18% at an annualized growth rates rather than scaled changes (Table 3); the rate. Excluding this quarter from the sample has no estimates of 2. are very slightly higher in this table, but important effects on the estimates of i..We conclude the general pattern is the same. We also obtain similar that the empirical results of Hall (1978), Flavin (1981), results when we use total consumption rather than con- and Nelson (1987), which use the 1950: 1 observation, sumption of nondurables and services and when we use should be treated with caution. the Blinder-Deaton (1985) data on consumption of Finally, we note that a split of the 1953-1985 sample nondurables and services and total disposable income in 1969: 2 gives similar estimates of i in both subsamples or disposable labor income. The distinction between but much greater statistical significance in the second these two income concepts seems to be unimportant subsample. As discussed in Campbell and Mankiw empirically. (1987), income growth is close to unpredictable in the Our estimates of /. are more sensitive to sample pe- first subsample. riod. As reported in detail in the National Bureau of Economic Research working paper version of this ar- 4. MONTE CARL0 RESULTS ticle (Campbell and Mankiw 1987), the addition of the four years 1949-1952 to the sample period makes the The evidence in Section 3 suggests that postwar U.S. estimates of i.small and insignificant, but this is entirely data can reject the permanent-income hypothesis. In due to one quarter, 1950: 1, in which there was a large this section, we use Monte Carlo methods to examine payment of National Service Life Insurance (NSLI) the small-sample distribution of our test statistics. The benefits to World War I1 veterans. Disposable income problem of small-sample bias has been a serious one in grew 6.596, or 26% at an annualized rate, in that tests of the permanent-income hypothesis (Mankiw and quarter. It seems inappropriate to treat this episode as Shapiro 1985), and it can be particularly dangerous being generated by the same time series process as the when using IV methods (Nelson and Startz 1988). Campbell and Mankiw: Permanent Income, Current Income, and Consumption 273

Table 3. Basic Permanent-Income Model (logs)

OLS regressions on z Instruments i estimate Test of Row [z) Ac equation Ay equation (standard error) restrictions None (OLS)

NOTE Thts table reports ~nstrumental-var~ablesesttmates of a log verslon of Equat~on(1) In the text Ac = ir + /Ay uslng Instrumental var~ablesz Lowercase letters denote log var~ables The statlstlcs In columns 3 and 4 are adjusted R2 values from OLS regresstons of Ac and lyon z [Eq (2) In the text] and slgnlflcance levels for tests of the hypothes~sthat all coeff~c~entsexcept the constant are 0 (In parentheses) The statlstlcs In column 5 are the ~nstrumental-variables estlmate of / wlth an asymptotic standard error (In parentheses) Column 6 glves the adjusted R2 from a regression of the IV restdual onto the Instruments and the s~gn~ftcancelevel for a Wald test of the over~dent~fy~ngrestrlctlons of the IV model All standard errors and test statlstlcs are heteroscedastlclty and autocorrelat~onconsistent

Monte Carlo experiments can protect us from excessive 7. The standard deviation of the log consumption- reliance on asymptotic distribution theory. income ratio (.0236 in the data) is more than twice the If our Monte Carlo experiment is to be convincing, standard deviation of income growth, and this ratio we need to use a data-generating process that both follows a first-order autoregressive [AR(l)] process obeys the restrictions of our model and matches the with a coefficient between .9 and .95. moments of the U.S. consumption and income data. 8. The contemporaneous correlation of income We have chosen to match the moments of the data growth and consumption growth is .58. (In combination measured in logs, since these are more familiar to most with property 6, this gives an OLS regression coefficient readers and are independent of the units of measure- of .33 when consumption growth is regressed on income ment. Our objective then is to generate data with the growth.) following properties: To obtain properties 1-5, we use a continuous-time 1. Log income and consumption are each first-order model first developed by Campbell and Kyle (1988). integrated processes. Details are given in the Appendix. Labor income and 2. The log consumption-income ratio is stationary, an information variable evolve according to a linear so consumption and income are cointegrated. vector-diffusion process with constant innovation vari- 3. Growth rates of income are not useful for fore- ance; this process is restricted in such a way that the casting growth rates of income or consumption two or univariate process for labor income is a Brownian mo- more quarters ahead. (This is an exaggeration of the tion (a continuous-time random walk). Consumers ob- properties of the actual data; in fact, income growth serve the information variable, however, and can rates do help to forecast income growth rates, but with therefore forecast future changes in labor income. A a very modest R', as shown in Table 3.) fraction i. of consumption is set equal to current income, 4. Growth rates of consumption and the log con- while a fraction (1 - i.) obeys a continuous-time version sumption-income ratio do forecast growth rates of in- of Flavin's (1981) permanent-income model. Total in- come two or more quarters ahead (with an R' of 5% come is the sum of labor income and endogenous capital to 15%). income. Observed variables are time averages of the 5. When we impose the permanent-income hypoth- underlying continuous processes. esis, the growth rate of consumption is not forecastable To obtain properties 6-8, we calibrate the free pa- by any variables two or more quarters ahead. rameters of the model. There are five in total but one 6. The standard deviation of consumption growth determines the overall variability of consumption and (.0053 in our data) is about half the standard deviation income, one is the coefficient i.,which we set equal to of income growth (.0094 in our data). 0 or .5, and one is the interest rate, which we fix at .01 274 Journal of Business & Economic Statistics, July 1990 per quarter; this leaves two free parameters to match experiment using the data-generating processes sum- relative standard deviations, cross-correlations, and au- marized in Table 4. For each process, we generated tocorrelations. 1,000 time series of length 125 and applied the econ- Having chosen the parameters of the model, we use ometric methods of Tables 2 and 3. The instrument sets the methods of Campbell and Kyle (1988) to calculate used are lags 2-4 of income growth, lags 2-6 of income the moments of the implied processes for time-averaged growth, lags 2-4 of consumption growth, lags 2-6 of consumption and income. Table 4 compares some im- consumption growth, and the lag 2 consumption-in- portant moments of the actual data with those gener- come ratio plus lags 2-4 of consumption and income ated by versions of the continuous-time model setting growth. For each instrument set, we report the mean 3. = 0 or .5. adjusted R' statistics for the consumption and income It is striking that the model in which 2 = 0 (artificial forecasting equations, and the mean IV estimate of 2. data set 1) is unable to account for the low standard with the mean standard error, the rejection rates for deviation of consumption growth, whereas the model nominal 5% and 1% OLS and instrumental-variables in which /. = .5 (artificial data set 2) matches this feature tests of the permanent-income hypothesis, and the em- of the data quite well. This is evidence that the presence pirical 5% and 1% critical values of the IV test. of current-income consumers can account for the find- In panel A of Table 5, the true value of 3. is 0; the ings of Campbell and Deaton (1989). The intuition for permanent-income hypothesis holds. The instruments this result was given by Flavin (1988). When consumers used in the first two rows have no forecasting power have superior information about the future path of in- for income, and the result is a severe upward bias in come, the revision in permanent income and the change the IV estimate of 2. The mean estimates are close to in current income are less than perfectly correlated. the mean OLS estimate of .405. In the remaining rows, This means that a weighted average of the two can be the instruments do forecast income and the bias in the less variable than either one taken alone. The model IV estimator is much smaller. This confirms the theo- in which 2 = .5 makes aggregate consumption a "di- retical analysis of Nelson and Startz (1988). versified portfolio" of the consumption of two types of The rejection rates of the IV tests in panel A are not agents, and this reduces its variability. too far from their theoretical values when three instru- In artificial data set 2, this effect is particularly pow- ments are used, but they increase with the number of erful because the revision in permanent income and the instruments. When seven instruments are used in the change in current income are actually negatively cor- final row, the true size of a nominal 5% test is almost related. The negative correlation also generates down- lo%, whereas the true size of a nominal 1% test is over ward bias in the OLS estimate of /. in Equation (1). 4%. Overall, the table suggests that the IV procedure More important for our present purposes, both ver- should be used only when the instruments have some sions of the continuous-time model have the property forecasting power for the right-side variable and that that lagged income-growth rates do not forecast future one should keep the number of instruments to a min- income-growth rates and are therefore invalid instru- imum. ments; lagged consumption-growth rates and consump- Panel B dramatically confirms the dangers of using tion-income ratios, by contrast, do have forecast power the IV procedure when the instruments do not forecast for income growth (and consumption growth when /. = income. In this panel, the true value of 2 is not 0 but .5). .5. In the first two rows, however, the instruments are In Table 5, we report the results of a Monte Carlo lagged income-growth rates, which forecast neither con- sumption nor income and do not identify ).. Neverthe- Table 4. Calibration of a Monte Carlo Experiment less, the IV test using three income instruments rejects the hypothesis 3. = 0 at the 5% level 25% of the time Artificial Artificial Actual data set 1 data set 2 and at the 1% level 10% of the time and the situation Statistic data (true i = 0) (true j. = .5) is even worse with five income instruments. In the remaining rows of panel B, the instruments do ~(AY,) ,009 ,010 ,010 ~(Ac,) ,005 ,010 ,005 forecast income growth, and the advantage of the IV Corr(Ay,, Ad .58 .42 .66 procedure becomes more apparent. The IV estimator Corr(Ay,, AY,.,) .06 .24 .24 has only a small downward bias. The IV method tests .25 .35 Corr(Ac,, Ac,.,) .16 the permanent-income model against a one-dimen- Corr(Ay1,AY,.,) - .18-.19 0 0 Corr(Ac,, AY,.,) - .la-.13 o o sional alternative, whereas the OLS test is against a Corr(Ay1,Ac,.,) - .16-.23 .07-.29 .lo-.13 multidimensional alternative. The IV test therefore has Corr(Ac,, Ac,.,) - .23-.20 0 .09-.13 more power against this particular alternative, and it ~(cr- ~ 1 ) ,024 ,018 ,021 rejects far more often than the OLS test. This is true Corr (c, - y,, c,., - y,.,) .95 .82 .94 whether one uses the theoretical critical values or the NOTE. Ai-~flclaldata are generated from the model described In the Appendix. Artlflclal empirical critical values from panel A. data set 1 has parameters, = 0, = 3, and p = .75 Art~flclaldata set 2 has parameters Our Monte Carlo results point to the same conclusion ,. ,. = .5,,ii = .I, and p = 1.5. The tlme Index 1 runs from 2 to 6 In the rows where 1 IS not speclfted. as the theoretical analysis of Nelson and Startz (1988). Campbell and Mankiw: Permanent Income, Current Income, and Consumption 275

Table 5. Monte Car10 Results

- Mean adjusted Rejection Empirical R2statistics probabilities critical values Instrument Ac AY Mean i OLS 5% IV 5% IV 5% set (z) on z on z estimate (OLS lo/') (IV 1%) (IV 1%)

A. Artificial data set 1 (true i = 0) None (OLS) ,405

Ay,, (1 = 2 - 4) ,001 ,001 ,399 ,052 (1.014) (.017) Ay,, (I = 2 - 6) ,001 ,000 ,396 ,066 (.558) (.011) ,039 ,034 ,063 (.555) (.016) Ac,., (i = 2 - 6) ,000 ,042 ,105 ,065 (.379) (.013) Ay,., (i = 2 - 4) .005 ,146 ,058 ,073 6c,., (i = 2 - 4) (.212) (.016) Cf-2 - Yf.2

6.Artificial data set 2 (true i = 5) None (OLS)

Ay,, (i = 2 - 4) ,002

Ay,., (i = 2 - 6) ,003

Ac,., (i = 2 - 4) ,018

Ac,., (i = 2 - 6) ,020

Ayi.,(i=2-4) ,131 Ac,., (i = 2 - 4) ci.2 - Yl-2

NOTE Thts table reports the results of a Monte Carlo s~mulat~onwlth 1 000 artlftctal data sets of length 125 The data were generated from the model descr~bedIn the Append~x In art~ftctaldata set 1 (panel A) true / = 0 and In artlf~claldata set 2 (panel B) true ,= 5 Columns 2 and 3 glve the emplrlcal mean adjusted R2 when Icand Iyare regressed on the Instruments z Column 4 glves the emplrlcal mean Instrumental var~ablesestlmate of I In the equatlon Jc = /I + ,Iy and the emptrlcal mean asymptotic standard error Column 5 glves the emplrlcal rejectton probab~llttesfor a 5% (1%) test of the hypothests that all coefflclents are 0 when IcIS regressed on z Column 6 glves the emplrlcal relectlon probabtl~tlesfor a 5% (1%) test of the hypothes~sthat, = 0 Column 7 in panel A gtves the emplrlcal 5% and 1% cr~t~calvalues for the t statlsttc on , The IV test should be used only with a moderate num- rate is constant. Any rejection of the theory might be ber of instruments that have adequate forecasting attributable to the failure of this assumption. For ex- power for income. Our regressions, however, seem to ample, Michener (1984) showed how variation through satisfy the conditions for reasonable small-sample be- time in the real interest rate can make consumption havior of the IV test. Table 5 suggests that the results appear excessively sensitive to income, even though of Tables 2 and 3 cannot be explained by small-sample individuals intertemporally optimize in the absence of bias. This is not surprising, given the rule of thumb of borrowing constraints. It is therefore important to ex- Nelson and Startz (1988) that with a single instrument amine whether the departure from the theory docu- the R2 for income growth should exceed 21T, or .016 mented previously is an artifact of the assumed con- for our data. In Tables 2 and 3. we have multiple in- stancy of the real interest rate. struments, so this condition is not directly applicable, The generalization of the consumer's Euler equation but it is encouraging to note that our adjusted R2sta- to allow for changes in the real interest rate is now well tistics are usually well above .02 and range up to .11. known (Grossman and Shiller 1981: Hall 1988; Hansen and Singleton 1983;Mankiw 1981.) The log-linear ver- 5. GENERALIZATIONS OF THE PERMANENT- ,ion of the Euler equation is INCOME HYPOTHESIS

In this section, we examine whether generalizations Ac, = ,u + (l/a)r, + E,. (3) of the permanent-income hypothesis, along some di- mension, can explain the rejection of the simple ~~odel where r, is the real interest rate contemporaneous with in Section 3. Ac, and, as before, the error term E, may be correlated with r, but is uncorrelated with lagged-variables. Ac- 5.1 Changes in the Real Interest Rate cording to (3), high ex ante real interest rates should Hall's (1978) random walk theorem for consumption be associated wi& rapid growth of consumption. If rests on the crucial assumption that the real interest higher income growth is associated with higher real in- 276 Journal of Business & Economic Statistics, July 1990

terest rates, this could explain the deviation from the 5.2 Nonseparabilities in the Utility Function permanent-income hypothesis documented previously. The random walk theorem for consumption will also To examine this possibility. we consider a more gen- fail if consumption is not separable in the utility function era1 model in which a fraction 1. of income goes to from other goods. With constant real interest rates, the individuals who consume their current income and the marginal utility of consumption is a martingale even remainder goes to individuals who satisfy the general under nonseparability; that is, it is still true that Euler equation (3). We estimate by IV

Ac, = p + iAy, + Or, + E,,

where 0 = (1 - %)/a.We thus include the actual income for some constant g. Yet predictable changes in the growth and the ex post real interest rate in the equation other good, X, must lead to predictable changes in con- but instrument using twice-lagged variables. The nom- sumption to maintain the martingale property of mar- inal interest rate we use is the average three-month ginal utility. If changes in X are correlated with changes treasury-bill rate over the quarter, the price index is the in income, nonseparability could in principle explain deflator for consumer nondurables and services, and the apparent excess sensitivity of consumption to in- we assume that there is a 30% marginal tax rate on come documented in Section 3. interest. (We obtained similar results when we assumed We test for nonseparability in a very simple way. We a marginal tax rate of 0.) The results for two alternative include the change in log X as an additional regressor instrument sets are in panel A of Table 6. in our equation. This functional form can be formally We find no evidence that the ex ante real interest justified if the utility function is Cobb-Douglas (Bean rate is associated with the growth rate of consumption. 1986) or a log-linear approximation to a more general The coefficient on the real interest rate is consistently specification. As before, we estimate the equation using less than its standard error. Moreover, the coefficient twice-lagged instrumental variables. on current income remains substantively and statisti- Various nonseparabilities have been proposed. cally significant. In contrast to the suggestion of Mich- Mankiw et al. (1985) and Eichenbaum et al. (1988) ener (1984). the excess sensitivity of consumption to considered nonseparability between consumption and income cannot be explained by fluctuations in the real labor supply. In panel B of Table 6, we include the interest rate. change in log labor supply as a right-side variable with

Table 6. Alternative Versions of the Permanent-Income Model

OLS regressions on z / 0 Instruments r (standard (standard Test of Row (2) Ac Ay or Ax error) error) restnctions

A. Real interest rates ,046 ,049 .467 ,451 (.018) (.004) (.OOO) (.log) ,077 ,026 ,448 ,668 (.OOO) (.010) (.OOO) (.173) B. Labor supply .029 ,079 ,221 ,364 (.031) (.001) (.OOO) (.200) ,086 ,062 ,150 ,497 (.OOO) (.OOO) (.OOO) (.212) C. Durable goods ,026 ,046 ,753 .258 (.136) (.012) (.OOO) (.143) ,115 .053 .727 ,590 (.OOO) (.001) (.OOO) (.137) D. Government spending

- ,001 ,043 ,046 .357 (.185) (.008) (.004) (.133) ,040 ,013 .037 ,664 (.001) (.016) (.016) (.170)

NOTE Panel A of th~stable reports ~nstrumental-var~ablesestlmates of Equat~on(6) In the text Ac = 11 - ,Ay + Or uslng instrumental var~ablesz The other panels report ~nstrumental-var~ablesestlmates when Ac = jl + ,Ay - ilAx where x IS labor supply durable goods or government spendtng The stattsttcs In columns 3, 4 and 5 are adlusted R2values from OLS regresstons of Ac, Ay, and r or Ax and Z and s~gn~f~cancelevels for tests of the hypothesis that all coeff~c~entsexcept the constant are 0 (In parentheses) The statlstlcs In columns 6 and 7 are the ~nstrumental-var~ablesestlmates of ,and 0, w~thasymptotic standard errors (in parentheses) Column 8 glves the adlusted R2 from a regresslon of the IV res~dualonto the Instruments and the slgnlflcance level lor a Wald test of the overldentlfylng restrlctlons of the IV model All standard errors and test statlstlcs are heteroscedastlclty and autocorrelatlon consistent Campbell and Mankiw: Permanent Income, Current Income, and Consumption 277 coefficient 0 [i.e., we estimate a version of Eq. (4) with plained by a model in the spirit of Flavin (1981), in the change in labor supply replacing the real interest which a fraction i of income goes to individuals who rate]. Labor supply is measured as per capita man-hours consume their current income rather than their per- in nonagricultural establishments. The results suggest manent income. This more general model is not sta- no important nonseparability between consumption and tistically rejected. Our estimates suggest that iis labor supply. Even though there is substantial predict- approximately .5, indicating a substantial departure able variation in the quantity of labor supplied, it is from the permanent-income hypothesis. apparently not associated with predictable changes in 3. The result that consumption tracks income too consumption. closely cannot be explained by the time-averaged nature Bernanke (1985) and Startz (1986) proposed that the of the data, by short delays in publication of aggregate marginal utility of nondurable goods may be affected statistics, or by partial durability of goods labeled "non- by the stock of consumer durable goods. In panel C of durable'' in the National Income Accounts. Our test of Table 6, we enter this stock as the X variable. We first the permanent-income model is robust to all of these constructed an end-of-quarter stock series from the an- problems because, in common with Hall (1988) but in nual stock at the beginning of the sample period and contrast with much of the rest of the literature, we lag the series on consumer durable purchases, assuming a our instruments by two quarters instead of one. depreciation rate of 6% per quarter. We then measured 4. The large estimate of i cannot be explained by the stock of consumer durables during a quarter as the small-sample bias in our IV estimation. Although IV average of the end-of-quarter stock and the previous procedures are vulnerable to small-sample bias when end-of-quarter stock. (Other timing assumptions lead the instruments are poor forecasters of the independent to similar results.) We find substantial predictable variables, in our application we have quite good fore- changes in the stock of durables but no evidence that casting power for future income growth. A carefully these changes coincide with predictable changes in con- calibrated Monte Carlo experiment does not generate sumption. enough small-sample bias to account for our results. It is often suggested that changes in government pur- 5. Our results cannot be explained by appealing to chases of goods and services affect the marginal utility more general versions of the permanent-income hy- of private consumption (Aschauer 1985; Bailey 1971; pothesis. We have allowed for changes in the real in- Kormendi 1983). Indeed. Aschauer suggested that al- terest rate and for nonseparability in the utility function lowing for such an effect can save the consumption between consumption and other goods-labor supply, Euler equation from a statistical rejection. In panel D consumer durables, and government purchases-but of Table 6, we examine this possibility by entering,the these generalizations do not help the model. change in the log of total government purchases per capita as a right-side variable. Again, we find no evi- ACKNOWLEDGMENTS dence of nonseparability in the utility function. More- We are grateful to Alan Blinder, Neil Ericsson, Mar- over, the estimate of i.we obtain remains statistically jorie Flavin, David Wilcox, and anonymous referees and substantively significant. In contrast to Aschauer, for helpful comments on an earlier draft. We acknowl- we find that nonseparability between private and public edge financial support from the National Science Foun- purchases does not improve the performance of the dation and the John M. Olin Fellowship at the National consumption Euler equation. Bureau of Economic Research.

6. CONCLUSIONS APPENDIX: CONSTRUCTION OF A MONTE CARL0 EXPERIMENT Our analysis of U.S. postwar quarterly data leads us to the following conclusions: Following Campbell and Kyle (1988, model B), sup- pose that labor income yl(t) and an information variable 1. There is evidence against the implication of the x(t) evolve in continuous time as permanent-income hypothesis that changes in con- sumption are unforecastable. When the change in con- sumption (scaled by lagged income or measured in logs) is regressed on its own lags 2-6 in the 1953-1985 period, the null hypothesis that all the coefficients are 0can be rejected at the .l% level. Although the adjusted R' of this regression is small (less than 9%), a small R2should Here dz, and dz2 are the increments to mutually or- not be viewed as supportive of the permanent-income thogonal Brownian motions. Campbell and Kyle hypothesis, since the R2of the comparable regression showed that (A.l) implies that the univariate process for the change in disposable income is also small. Sim- for yl(t) is a Brownian motion, but the univariate pro- ilar results can be obtained using lagged nominal in- cess for x(t) is an Ornstein-Uhlenbeck [continuous-time terest rates as instruments. AR(l)] process that reverts to its zero mean at rate a. 2. The forecastability of consumption can be ex- Knowledge of yl(t) does not help to forecast x(t), but 278 Journal of Business & Economic Statistics, July 1990

x(t) does help to forecast yl(t). If E,denotes expectation Campbell. J. Y. (1987), "Does Saving Anticipate Declining Labor conditional on x(t) and yl(t), then Income? An Alternative Test of the Permanent Income Hypoth- esis," Econometrica. 55, 1249-1273. Campbell, J. Y., and Deaton, A. S. (1989), "Why Is Consumption e-"yl(t+s)ds = yl(t) + $x(t). (A.2) So Smooth?" Review of Economic Studies. 56. 357-374. I,:, I,:, Campbell. J. Y., and Kyle, A. S. (1988),"Smart-Money. Noise Trad- ing, and Stock Price Behavior," Technical Working Paper 71, Na- where $ = 1 - r/(r + a). tional Bureau of Economic Research, Cambridge, MA. We now define total income as y(t) = yl(t) + rw(t). Campbell, J. Y., and Mankiw, N. G. (1987), "Permanent Income, where the second term is capital income received from Current Income. and Consumption." Working Paper 2436, Na- real wealth w(t). We suppose that consumption is de- tional Bureau of Economic Research, Cambridge. MA. -(1989),"Consumption, Income, and Interest Rates: Reinter- termined by preting the Time Series Evidence," in NBER Macroeconomics An- nual, eds. 0.Blanchard and S. Fischer. Cambridge, MA: MIT Press, pp. 185-216. Christiano, L. J., Eichenbaum, M., and Marshall, D. (1987), "The Permanent Income Hypothesis Revisited," Research Department Working Paper 335. Federal Reserve Bank of Minneapolis. Davidson, J. E. H., and Hendry, D. F. (1981)."Interpreting Econ- = y(t) + (1 - I.)$x(t) from (A.2). (A.3) ometric Evidence: The Behavior of Consumers' Expenditure in the U.K.," European Economic Review, 16, 177-192. The evolution of real wealth is given by Davidson, J. E. H., Hendry, D. F., Srba, F., and Yeo, S. (1978), "Econometric Modelling of the Aggregate Time-Series Relation- ship Between Consumers' Expenditure and Income in the United One interesting feature of this model is that the in- Kingdom," Economic Journal, 88, 661-692. Deaton, A. S. (1987), "Life-Cycle Models of Consumption: Is the novation to current labor income dy(t) has covariance Evidence Consistent With the Theory?" in Advances in Econo- 02(l - $p) with the innovation to permanent income. metrics, Fifth World Congress (Vol. 2), ed. T. F. Bewley, Cam- If 4 and p are large enough (as they are in artificial data bridge. U.K.: Cambridge University Press, pp. 121-148. set 2), this covariance can be negative. OLS estimates DeLong. J. B.. and Summers, L. H. (1986)."The Changing Cyclical of I. are then downward biased, as they appear to be Variability of Economic Activity in the United States," in The American Business Cycle: Continuity and Change, ed. R. J. Gor- in the data. don. Chicago: University of Chicago Press. pp. 679-734. Equations (A.l)-(A.4) are written in terms of log Dornbusch, R., and Fischer, S. (1987), Macroeconomics (5th ed.). variables, even though the model appears to be appro- New York: McGraw-Hill. priate for levels. This can be justified either as a simple Eichenbaum, M. S., Hansen, L. P., and Singleton, K. J. (1988)."A way to generate artificial data that satisfy properties 1- Time Series Analysis of Representative Agent Models of Con- sumption and Leisure Choice Under Uncertainty." Quarterly Jour- 8 in the text or by using a log-linear approximation of nal of Economics. 103, 51-78. the budget constraint to state the permanent-income Engle, R. F., and Granger, C. W. J. (1987), "Co-Integration and model in log terms (Campbell and Mankiw 1989). Error-Correction: Representation, Estimation and Testing," Econ- Campbell and Kyle (1988) showed how to calculate ometric~,55, 251-276. the time series process for time averages of variables Fischer, S., and Merton, R. C. (1984). "Macroeconomics and Fi- nance: The Role of the Stock Market," in Carnegie-Rochester which obey (A.l)-(A.4). To calibrate our Monte Carlo Conference Series on Public Policy (Vol. 21),eds. K. 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Fiscal Policy and Aggregate Demand David Alan Aschauer The American Economic Review, Vol. 75, No. 1. (Mar., 1985), pp. 117-127. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28198503%2975%3A1%3C117%3AFPAAD%3E2.0.CO%3B2-B

The Estimation of "Surprise" Models and the "Surprise" Consumption Function Charles R. Bean The Review of Economic Studies, Vol. 53, No. 4, Econometrics Special Issue. (Aug., 1986), pp. 497-516. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28198608%2953%3A4%3C497%3ATEO%22MA%3E2.0.CO%3B2-T

The Time Series Consumption Function Revisited Alan S. Blinder; Angus Deaton; Robert E. Hall; R. Glenn Hubbard Brookings Papers on Economic Activity, Vol. 1985, No. 2. (1985), pp. 465-521. Stable URL: http://links.jstor.org/sici?sici=0007-2303%281985%291985%3A2%3C465%3ATTSCFR%3E2.0.CO%3B2-6 http://www.jstor.org

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Does Saving Anticipate Declining Labor Income? An Alternative Test of the Permanent Income Hypothesis John Y. Campbell Econometrica, Vol. 55, No. 6. (Nov., 1987), pp. 1249-1273. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198711%2955%3A6%3C1249%3ADSADLI%3E2.0.CO%3B2-J

Why is Consumption So Smooth? John Campbell; Angus Deaton The Review of Economic Studies, Vol. 56, No. 3. (Jul., 1989), pp. 357-373. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28198907%2956%3A3%3C357%3AWICSS%3E2.0.CO%3B2-V

Econometric Modelling of the Aggregate Time-Series Relationship Between Consumers' Expenditure and Income in the United Kingdom James E. H. Davidson; David F. Hendry; Frank Srba; Stephen Yeo The Economic Journal, Vol. 88, No. 352. (Dec., 1978), pp. 661-692. Stable URL: http://links.jstor.org/sici?sici=0013-0133%28197812%2988%3A352%3C661%3AEMOTAT%3E2.0.CO%3B2-J

A Time Series Analysis of Representative Agent Models of Consumption and Leisure Choice under Uncertainty Martin S. Eichenbaum; Lars Peter Hansen; Kenneth J. Singleton The Quarterly Journal of Economics, Vol. 103, No. 1. (Feb., 1988), pp. 51-78. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28198802%29103%3A1%3C51%3AATSAOR%3E2.0.CO%3B2-V

Co-Integration and Error Correction: Representation, Estimation, and Testing Robert F. Engle; C. W. J. Granger Econometrica, Vol. 55, No. 2. (Mar., 1987), pp. 251-276. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198703%2955%3A2%3C251%3ACAECRE%3E2.0.CO%3B2-T

The Adjustment of Consumption to Changing Expectations About Future Income Marjorie A. Flavin The Journal of Political Economy, Vol. 89, No. 5. (Oct., 1981), pp. 974-1009. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198110%2989%3A5%3C974%3ATAOCTC%3E2.0.CO%3B2-X http://www.jstor.org

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The Determinants of the Variability of Stock Market Prices Sanford J. Grossman; Robert J. Shiller The American Economic Review, Vol. 71, No. 2, Papers and Proceedings of the Ninety-Third Annual Meeting of the American Economic Association. (May, 1981), pp. 222-227. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28198105%2971%3A2%3C222%3ATDOTVO%3E2.0.CO%3B2-8

Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence Robert E. Hall The Journal of Political Economy, Vol. 86, No. 6. (Dec., 1978), pp. 971-987. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197812%2986%3A6%3C971%3ASIOTLC%3E2.0.CO%3B2-K

Intertemporal Substitution in Consumption Robert E. Hall The Journal of Political Economy, Vol. 96, No. 2. (Apr., 1988), pp. 339-357. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198804%2996%3A2%3C339%3AISIC%3E2.0.CO%3B2-8

The Sensitivity of Consumption to Transitory Income: Estimates from Panel Data on Households Robert E. Hall; Frederic S. Mishkin Econometrica, Vol. 50, No. 2. (Mar., 1982), pp. 461-481. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198203%2950%3A2%3C461%3ATSOCTT%3E2.0.CO%3B2-R

Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns Lars Peter Hansen; Kenneth J. Singleton The Journal of Political Economy, Vol. 91, No. 2. (Apr., 1983), pp. 249-265. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198304%2991%3A2%3C249%3ASCRAAT%3E2.0.CO%3B2-1

The Permanent Income Hypothesis: Estimation and Testing by Instrumental Variables Fumio Hayashi The Journal of Political Economy, Vol. 90, No. 5. (Oct., 1982), pp. 895-916. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198210%2990%3A5%3C895%3ATPIHEA%3E2.0.CO%3B2-Y http://www.jstor.org

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Government Debt, Government Spending, and Private Sector Behavior Roger C. Kormendi The American Economic Review, Vol. 73, No. 5. (Dec., 1983), pp. 994-1010. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28198312%2973%3A5%3C994%3AGDGSAP%3E2.0.CO%3B2-H

Money, Real Interest Rates, and Output: A Reinterpretation of Postwar U.S. Data Robert B. Litterman; Laurence Weiss Econometrica, Vol. 53, No. 1. (Jan., 1985), pp. 129-156. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198501%2953%3A1%3C129%3AMRIRAO%3E2.0.CO%3B2-Y

Intertemporal Substitution in Macroeconomics N. Gregory Mankiw; Julio J. Rotemberg; Lawrence H. Summers The Quarterly Journal of Economics, Vol. 100, No. 1. (Feb., 1985), pp. 225-251. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28198502%29100%3A1%3C225%3AISIM%3E2.0.CO%3B2-2

A Reappraisal of Recent Tests of the Permanent Income Hypothesis Charles R. Nelson The Journal of Political Economy, Vol. 95, No. 3. (Jun., 1987), pp. 641-646. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198706%2995%3A3%3C641%3AARORTO%3E2.0.CO%3B2-E

Comparison of Interwar and Postwar Business Cycles: Monetarism Reconsidered Christopher A. Sims The American Economic Review, Vol. 70, No. 2, Papers and Proceedings of the Ninety-Second Annual Meeting of the American Economic Association. (May, 1980), pp. 250-257. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28198005%2970%3A2%3C250%3ACOIAPB%3E2.0.CO%3B2-N

Note on the Correlation of First Differences of Averages in a Random Chain Holbrook Working Econometrica, Vol. 28, No. 4. (Oct., 1960), pp. 916-918. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28196010%2928%3A4%3C916%3ANOTCOF%3E2.0.CO%3B2-X