2.2.1 Approximation of the Euler equation

Blundell et al. (2008b) consider the consumption problem faced by household i of age a in period t. Assuming that preferences are of the CRRA form, the objective is to choose a path for consumption C so as to:

(6)

where Z_i,a+j,t+j incorporates taste shifters (such as age, household composition, etc.), and we denote with E_a (.) = E(.|Ω_i,a,t). Maximization of (6) is subject to the budget constraint, which in the self-insurance model assumes individuals have access to a risk free bond with real return r

(7)

(8)

with A_i,a,t given. Blundell et al. (2008b) set the retirement age after which labor income falls to zero at L, assumed known and certain, and the end of the life cycle at age A. They assume that there is no uncertainty about the date of death. With budget constraint (7), optimal consumption choices can be described by the Euler equation (assuming for simplicity that there is no preference heterogeneity, or υ_a = 0):

(9)

As it is, Eq. (9) is not useful for empirical purposes. Blundell et al. (2008b) show that the Euler equation can be approximated as follows:

where η_i,a,t is a consumption shock with E_a−1(η_i,a,t) = 0, captures any slope in the consumption path due to interest rates, impatience or precautionary savings and the error in the approximation is .¹⁷ Suppose that any idiosyncratic component to this gradient to the consumption path can be adequately picked up by a vector of deterministic characteristics and a stochastic individual element η_i,a

Assume log income is

(10)

(11)

where represent observable characteristics influencing the growth of income. Income growth can be written as:

The (ex-post) intertemporal budget constraint is

where A is the age of death and L is the retirement age. Applying the approximation above and taking differences in expectations gives

where π_a is an annuitization factor, is the share of future labor income in current human and financial wealth, and the error of the approximation is . Then¹⁸

(12)

with a similar order of approximation error.¹⁹ The random term Ξ_i,a,t can be interpreted as the innovation to higher moments of the income process.²⁰ As we shall see, Meghir and Pistaferri (2004) find evidence of this using PSID data.

The interpretation of the impact of income shocks on consumption growth in the PIH model with CRRA preferences is straightforward. For individuals a long time from the end of their life with the value of current financial assets small relative to remaining future labor income, , and permanent shocks pass through more or less completely into consumption whereas transitory shocks are (almost) completely insured against through saving. Precautionary saving can provide effective self-insurance against permanent shocks only if the stock of assets built up is large relative to future labor income, which is to say Ξ_i,a,t is appreciably smaller than unity, in which case there will also be some smoothing of permanent shocks through self-insurance.

The most important feature of the approximation approach is to show that the effect of an income shock on consumption depends not only on the persistence of the shock and the planning horizon (as in the LC-PIH case with quadratic preferences), but also on preference parameters. Ceteris paribus, the consumption of more prudent households will respond less to income shocks. The reason is that they can use their accumulated stock of precautionary wealth to smooth the impact of the shocks (for which they were saving precautiously against in the first place). Simulation results (below) confirm this basic intuition.

2.2.2 Kaplan and Violante

Kaplan and Violante (2009) investigate the amount of consumption insurance present in a life cycle version of the standard incomplete markets model with heterogenous agents (e.g., Rios-Rull, 1996; Huggett, 1996). Kaplan and Violante’s setup differs from that in Blundell et al. (2008b; BPP) by adding the uncertainty component μ_α to life expectancy, and by omitting the taste shifters from the utility function. μ_α is the probability of dying at age a. It is set to 0 for all a < L (the known retirement age) and it is greater than 0 for L ≤ a ≤ A. Their model also differs from BPP by specifying a realistic social security system. Two baseline setups are investigated—a natural borrowing constraint setup (henceforth NBC), in which consumers are only constrained by their budget constraint, and a zero borrowing constraint setup (henceforth ZBC), in which consumers have to maintain non-negative assets at all ages. The income process is similar to BPP.²¹ Part of Kaplan and Violante’s analysis is designed to check whether the amount of insurance predicted by the Bewley model can be consistently estimated using the identification strategy proposed by BPP and whether BPP’s estimates using PSID and CEX data conform to values obtained from calibrating their theoretical model.

Kaplan and Violante (2009) calibrate their model to match the US data. Survival rates are obtained from the NCHS, the intertemporal discount rate is calibrated to match a wealth-income ratio of 2.5, the permanent shock parameters and the variance of the initial draw of the process) are calibrated to match PSID data and the variance of the transitory shock is set to the 1990–1992 BPP point estimate (0.05). The Kaplan and Violante (2009) model is solved numerically. This allows for the calculation of both the “true”²² and the BPP estimators of the “partial insurance parameters” (the response of consumption to permanent and transitory income shocks).

Figure 2 is reproduced from Kaplan and Violante (2009).²³ It plots the theoretical marginal propensity to consume for the transitory shocks (upper panels) and the permanent shocks (lower panels) against age (continuous line) and those obtained using BPP’s identification methodology (dashed line). The left panels refer to the NBC environment; the right panels to the ZBC environment. A number of interesting findings emerge. First, in the NBC environment the MPC with respect to transitory shocks is fairly low throughout the life cycle, and similarly to what is shown in Fig. 1, increases over the life cycle due to reduced planning horizon effect. The life cycle average MPC is 0.06. Second, there is considerable insurance also against permanent shock, which increases over the life cycle due to the ability to use the accumulated wealth to smooth these shocks. The life cycle average MPC is 0.77, well below the MPC of 1 predicted by the infinite horizon PIH model.²⁴ Third, the ZBC environment affects only the ability to insure transitory shocks (which depend on having access to loans), but not the ability to insure permanent shocks (which depend on having access to a storage technology, and hence it is not affected by credit restrictions). Fourth, the performance of the BPP estimators is remarkably good. Only in the case of the ZBC environment and a permanent shock does the BPP estimator display an upward bias, and even in that case only very early in the life cycle. According to KV the source of the bias is the failure of the orthogonality condition used by BPP for agents close to the borrowing constraint. It is worth noting that the ZBC environment is somewhat extreme as it assumes no unsecured borrowing. Finally, KV compare the average MPCs obtained in their model (0.06 and 0.77) with the actual estimates obtained by BPP using actual data. As we shall see, BPP find an estimate of the MPC with respect to permanent shocks of 0.64 (s.e. 0.09) and an estimate of the MPC with respect to transitory shocks of 0.05 (s.e. 0.04). Clearly, the “theoretical” MPCs found by KV lie well in the confidence interval of BPP’s estimates. One thing that seems not to be borne out in the data is that theoretically the degree of smoothing of permanent shocks should be strictly increasing and convex with age, while BPP report an increasing amount of insurance with age as a non-significant finding.²⁵ As discussed by Kaplan and Violante (2009), the theoretical pattern of the smoothing coefficients is the result of two forces: a wealth composition effect and a horizon effect. The increase in wealth over the life cycle due to precautionary and retirement motives means that agents are better insured against shocks. As the horizon shortens, the effect of permanent shock resembles increasingly that of a transitory shock.

Given that the response of consumption to shocks of various nature is so different (and so relevant for policy in theory and practice), it is natural to turn to studies that analyze the nature and persistence of the income process.

3.1.1 A simple model of earnings dynamics

We start with the relatively simpler representation where the term a × f_i is excluded. Moreover we restrict the lag polynomials ©(L) and P(L): it is not generally possible to identify ©(L) and P(L) without any further restrictions. Thus we start with the typical specification used for example in MaCurdy (1982) and Abowd and Card (1989):

(15)

with m_i,a,t, ζ_i,a,t and ε_εi,a,t all being independently and identically distributed and where h_i reflects initial heterogeneity, which here persists forever through the random walk (a = 0 is the age of entry in the labor market, which may differ across groups due to different school leaving ages). Generally, as we will show, the existence of classical measurement error causes problems in the identification of the transitory shock process.

There are two principal motivations for the permanent/transitory decompositions: the first motivation draws from economics: the decomposition reflects well the original insights of Friedman (1957) by distinguishing how consumption can react to different types of income shock, while introducing uncertainty into the model.³⁰ The second is statistical: At least for the US and for the UK the variance of income increases over the life cycle (see Fig. 3, which uses consumption data from the CEX and income data from the PSID). This, together with the increasing life cycle variance of consumption points to a unit root in income, as we shall see below. Moreover, income growth (Δy_i,a,t) has limited serial correlation and behaves very much like an MA process of order 2 or three: this property is delivered by the fact that all shocks above are assumed i.i.d. In our example growth in income has been restricted to an MA(2).³¹

Even in such a tight specification identification is not straightforward: as we will illustrate we cannot separately identify the parameter θ, the variance of the measurement error and the variance of the transitory shock. But first consider the identification of the variance of the permanent shock. Define unexplained earnings growth as:

(16)

Then the key moment condition for identifying the variance of the permanent shock is

(17)

where q is the order of the moving average process in the original levels equation; in our example q = 1. Hence, if we know the order of serial correlation of log income we can identify the variance of the permanent shock without any need to identify the variance of the measurement error or the parameters of the MA process. Indeed, in the absence of a permanent shock the moment in (17) will be zero, which offers a way of testing for the presence of a permanent component conditional on knowing the order of the MA process. If the order of the MA process is one in the levels, then to implement this we will need at least six individual-level observations to construct this moment. The moment is then averaged over individuals and the relevant asymptotic theory for inference is one that relies on a large number of individuals N.

At this point we need to mention two potential complications with the econometrics. First, when carrying out inference we have to take into account that y_i,a,t has been constructed using the pre-estimated parameters d_t and β in Eq. (13). Correcting the standard errors for this generated regressor problem is relatively simple to do and can be done either analytically, based on the delta method, or just by using the bootstrap. Second, as said above, to estimate such a model we may have to rely on panel data where individuals have been followed for the necessary minimum number of periods/years (6 in our example); this means that our results may be biased due to endogenous attrition. In practice any adjustment for this is going to be extremely hard to do because we usually do not observe variables that can adequately explain attrition and at the same time do not explain earnings. Administrative data may offer a promising alternative to relying on attrition-prone panel data.

The order of the MA process for υ_i,a,t will not be known in practice and it has to be estimated. This can be done by estimating the autocovariance structure of g_i,a,t and deciding a priori on the suitable criterion for judging whether they should be taken as zero. One approach followed in practice is to use the t-statistic or the F-statistics for higher order autocovariances. However, we need to recognize that given an estimate of q the analysis that follows is conditional on that estimate of q, which in turn can affect inference, particularly for the importance of the variance of the permanent effect .

3.1.2 Estimating and identifying the properties of the transitory shock

The next issue is the identification of the parameters of the moving average process of the transitory shock and those of measurement error. It turns out that the model is underidentified, which is not surprising: in our example we need to estimate three parameters, namely the variance of the transitory shock , the MA coefficient θ and the variance of the measurement error .³² To illustrate the underidentification point suppose that |θ| < 1 and assume that the measurement error is independently and identically distributed. We take as given that q = 1. Then the autocovariances of order higher than three will be zero, whatever the value of our unknown parameters, which is the root of the identification problem. The first and second order autocovariances imply

(18)

The sign of E(g_i,a,tg_{i,a–2,t−2}) defines the sign of θ. Taking the two variances as functions of the MA coefficient we note two points. First, declines and increases when θ declines in absolute value. Second, for sufficiently low values of |θ| the estimated variance of the measurement error may become negative. Given the sign of θ (defined by I in Eq. (18)) this fact defines a bound for the MA coefficient. Suppose for example that θ < 0, we have that , where is the negative value of θ that sets in (18) to zero. If θ was found to be positive the bounds would be in a positive range. The bounds on θ in turn define bounds on and .

An alternative empirical strategy is to rely on an external estimate of the variance of the measurement error, . Define the moments, adjusted for measurement error as:

where is available externally. The three moments above depend only on θ, and . We can then estimate these parameters using a Minimum Distance procedure.

Such external measures can sometimes be obtained through validation studies. For example, Bound and Krueger (1991) conduct a validation study of the CPS data on earnings and conclude that measurement error explains 35 percent of the overall variance of the rate of growth of earnings of males in the CPS. Bound et al. (1994) find a value of 26 percent using the PSID-Validation Study.³³

3.1.3 Estimating alternative income processes

Time varying impacts An alternative specification with very different implications is one where

(19)

where h_i is a fixed effect while υ_i,a,t follows some MA process and m_i,a,t is measurement error (see Holtz-Eakin et al., 1988). This process can be estimated by method of moments following a suitable transformation of the model. Define θ_t = d_t/d_t−1 and quasi-difference to obtain:

(20)

In this model the persistence of the shocks is captured by the autoregressive component of ln Y, which means that the effects of time varying characteristics are persistent to an extent. Given estimates of the levels equation in (20) the autocovariance structure of the residuals can be used to identify the properties of the error term d_tΔυ_i,a,t + m_i,a,t−θ_tm_{i,a−1,t−1}.

Alternatively, the fixed effect with the autoregressive component can be replaced by a random walk in a similar type of model. This could take the form

(21)

In this model p_i,a,t = p_{i,a−1,t−1} + ζ_i,a,t as before, but the shocks have a different effect depending on aggregate conditions. Given fixed T a linear regression in levels can provide estimates for dt, which can now be treated as known.

Now define θ_t = d_t/d_t−1 and consider the following transformation

(22)

The autocovariance structure of ln Y_i,a,t − θ_t ln Y_{i,a−1,t−1} can be used to estimate the variances of the shocks, very much like in the previous examples. We will not be able to identify separately the variance of the transitory shock from that of measurement error, just like before. In general, one can construct a number of variants of the above model but we will move on to another important specification, keeping from now on any macroeconomic effects additive.

It should be noted that (22) is a popular model among labor economists but not among macroeconomists. One reason is that it is hard to use in macro models—one needs to know the entire sequence of prices, address general equilibrium issues, etc.

Stochastic growth in earnings Now consider generalizing in a different way the income process and allow the residual income growth (16) to become

(23)

where the f_i is a fixed effect. The fundamental difference of this specification from the one presented before is that the income growth of a particular individual will be correlated over time. In the particular specification above, all theoretical autocovariances of order three or above will be equal to the variance of the fixed effect f_i. Consider starting with the null hypothesis that the model is of the form presented in (15) but with an unknown order for the MA process governing the transitory shock υ_i,a,t = Θ_q(L)ε_i,a,t. In practice we will have a panel data set containing some finite number of time series observations but a large number of individuals, which defines the maximum order of autocovariance that can be estimated. In the PSID these can be about 30 (using annual data). The pattern of empirical autocovariances consistent with (16) is one where they decline abruptly and become all insignificantly different from zero beyond that point. The pattern consistent with (23) is one where the autocovariances are never zero but after a point become all equal to each other, which is an estimate of the variance of f_i.

Evidence reported in MaCurdy (1982), Abowd and Card (1989), Topel and Ward (1992), Moffitt and Gottschalk (1994) and Meghir and Pistaferri (2004) and others all find similar results: Autocovariances decline in absolute value, they are statistically insignificant after the 1st or 2nd order, and have no clear tendency to be positive. They interpret this as evidence that there is no random growth term. Figure 4 uses PSID data and plot the second, third and fourth order autocovariances of earnings growth (with 95% confidence intervals) against calendar time. They confirm the findings in the literature: After the second lag no autocovariance is statistically significant for any of the years considered, and there are as many positive estimates as negative ones. In fact, there is no clear pattern in these estimates.

With a long enough panel and a large number of cross sectional observations we should be able to detect the difference between the two alternatives. However, there are a number of practical and theoretical difficulties. First, with the usual panel data, the higher order autocovariances are likely to be estimated based on a relatively low number of individuals. This, together with the fact that the residuals already contain noise from removing the estimated effects of characteristics such as age and even time effects will mean that higher order autocovariances are likely to be imprecisely estimated, even if the variance of f_i is indeed non-zero. Perhaps administrative data is one way round this, because we will be observing long run data on a large number of individuals. However, such data is not always available either because it is not organized in a usable way or because of confidentiality issues.

The other issue is that without a clearly articulated hypothesis we may not be able to distinguish among many possible alternatives, because we do not know the order of the MA process, q, or even if we should be using an MA or AR representation, or if the “permanent component” has a unit root or less. If we did, we could formulate a method of moments estimator and, subject to the constraints from the amount of years we observe, we could estimate our model and test our null hypothesis.

The practical identification problem is well illustrated by an argument in Guvenen (2009). Consider the possibility that the component we have been referring to as permanent, p_i,a,t, does not follow a random walk, but follows some stationary autoregressive process. In this case the increase in the variance over the life cycle will be captured by the term α × f_i. The theoretical autocovariances of g_i,a,t will never become exactly zero; they will start negative and gradually increase asymptotically to a positive number which will be the variance of f_i, say . Specifically if p_i,a,t = RP_{i,a−1,t−1} + ζ_i,a,t with |ρ| < 1, there is no other transitory stochastic component, and the variance of the initial draw of the permanent component is zero, the autocovariances of order k have the form

(24)

As ρ approaches one the autocovariances will approach . However, the autocovariance in (24) is the sum of a positive and a negative component. Guvenen (2009) has shown, based on simulations, that it is almost impossible in practice with the usual sample sizes to distinguish the implied pattern of the autocovariances from (24) from the one estimated from PSID data. The key problem with this is that the usual panel data that is available either follows individuals for a limited number of time periods, or suffers from severe attrition, which is probably not random, introducing biases. Thus, in practice it is very difficult to identify the nature of the income process without some prior assumptions and without combining information with another process, such as consumption or labor supply.

Haider and Solon (2006) provide a further illustration of how difficult it is to distinguish one model from the other. They are interested in the association between current and lifetime income. They write current log earnings as

y_i,a,t = h_i + af_i

and lifetime earnings as (approximately)

log V_i = r − log r + h_i + r⁻¹ f_i.

The slope of a regression of y_i,a,t onto log V_i is:

Hence, the model predicts that λ_a should increase linearly with age. In the absence of a random growth term , λ_a = 1 at all ages. Figure 5, reproduced from Haider and Solon (2006) shows that there is evidence of a linear growth in λ_a only early in the life cycle (up until age 35); however, between age 35 and age 50 there is no evidence of a linear growth in λ_a (if anything, there is evidence that λ_a declines and one fails to reject the hypothesis λ_a = 1); finally, after age 50, there is evidence of a decline in λ_a that does not square well with any random growth term in earnings.

Other enrichments/issues The literature has addressed many other interesting issues having to do with wage dynamics, which here we only mention in passing. First, the importance of firm or match effects. Matched employer-employee data could be used to address these issues, and indeed some papers have taken important steps in this direction (see Abowd et al., 1999; Postel-Vinay and Robin, 2002; Guiso et al., 2005).

A number of papers have remarked that wages fall dramatically at job displacement, generating so-called “scarring” effects (Jacobson et al., 1993; von Wachter et al., 2007). The nature of these scarring effects is still not very well understood. On the one hand, people may be paid lower wages after a spell of unemployment due to fast depreciation of their skills (Ljunqvist and Sargent, 1998). Another explanation could be loss of specific human capital that may be hard to immediately replace at a random firm upon re-entry (see Low et al., forthcoming).

3.1.4 The conditional variance of earnings

The typical empirical strategy followed in the precautionary savings literature, in the attempt to understand the role of risk in shaping household asset accumulation choices, typically proceeds in two steps. In the first step, risk is estimated from a univariate ARMA process for earnings (similar to one of those described earlier). Usually the variance of the residual is the assumed measure of risk. There are some variants of this typical strategy—for example, allowing for transitory and permanent income shocks. In the second step, the outcome of interest (assets, savings, or consumption growth) is regressed onto the measure of risk obtained in the first stage, or simulations are used to infer the importance of the precautionary motive for saving. Examples include Banks et al. (2001) and Zeldes (1989). In one of the earlier attempts to quantify the importance of the precautionary motive for saving, Caballero (1990) concluded −using estimates of risk from MaCurdy (1982)—that precautionary savings could explain about 60% of asset accumulation in the US.

A few recent papers have taken up the issue of risk measurement (i.e., modeling the conditional variance of earnings) in a more complex way. Here we comment primarily on Meghir and Pistaferri (2004).³⁴

Meghir and Pistaferri (2004) Returning to the model presented in Section 3.1.1 we can extend this by allowing the variances of the shocks to follow a dynamic structure with heterogeneity. A relatively simple possibility is to use ARCH(1) structures of the form

(25)

where E_t–1 (.) denotes an expectation conditional on information available at time t − 1. The parameters are all education-specific. Meghir and Pistaferri (2004) test whether they vary across education. The terms γ_t and ϕ_t are year effects which capture the way that the variance of the transitory and permanent shocks change over time, respectively. In the empirical analysis they also allow for life cycle effects. In this specification we can interpret the lagged shocks (ε_{i,a−1,t−1}, ζ_{i,a−1,t−1}) as reflecting the way current information is used to form revisions in expected risk. Hence it is a natural specification when thinking of consumption models which emphasize the role of the conditional variance in determining savings and consumption decisions.

The terms v¿ and ξ; are fixed effects that capture all those elements that are invariant over time and reflect long term occupational choices, etc. The latter reflects permanent variability of income due to factors unobserved by the econometrician. Such variability may in part have to do with the particular occupation or job that the individual has chosen. This variability will be known by the individuals when they make their occupational choices and hence it also reflects preferences. Whether this variability reflects permanent risk or not is of course another issue which is difficult to answer without explicitly modeling behavior.³⁵

As far as estimating the mean and variance process of earnings is concerned, this model does not require the explicit specification of the distribution of the shocks; moreover the possibility that higher order moments are heterogeneous and/or follow some kind of dynamic process is not excluded. In this sense it is very well suited for investigating some key properties of the income process. Indeed this is important, because as discussed earlier the properties of the variance of income have implications for consumption and savings.

However, this comes at a price: first, Meghir and Pistaferri (2004) need to impose linear separability of heterogeneity and dynamics in both the mean and the variance. This allows them to deal with the initial conditions problem without any instruments. Second, they do not have a complete model that would allow them to simulate consumption profiles. Hence the model must be completed by specifying the entire distribution.

Identification of the ARCH process If the shocks ε and ζ were observable it would be straightforward to estimate the parameters of the ARCH process in (25). However they are not. What we do observe (or can estimate) is g_i,a,t = Δm_i,a,t + (1+θL) Δε_i,a,t + ζ_i,a,t. To add to the complication we have already argued that θ is not point identified. Nevertheless the following two key moment conditions identify the parameters of the ARCH process, conditional on the unobserved heterogeneity (η and ξ):

(26)

The important point here is that it is sufficient to know the order of the MA process q.³⁶ We do not need to know the parameters themselves. The parameter θ that appears in (26) for the transitory shock is just absorbed by the time effects on the variance or the heterogeneity parameter. Hence measurement error, which prevents the identification of the MA process does not prevent identification of the properties of the variance, so long as such error is classical.

The moments above are conditional on unobserved heterogeneity; to complete identification we need to control for that. As the moment conditions demonstrate, estimating the parameters of the variances is akin to estimating a dynamic panel data model with additive fixed effects. Typically we should be guided in estimation by asymptotic arguments that rely on the number of individuals tending to infinity and the number of time periods being fixed and relatively short.

One consistent approach to estimation would be to use first differences to eliminate the heterogeneity and then use instruments dated t − 3 for the transitory shock and dated t − q − 4 for the permanent one. In this case the moment conditions become

(27)

where Δx_t = x_t − x_t−1. In practice, however, as Meghir and Pistaferri (2004) found out, lagged instruments suggested above may be only very weakly correlated with the entities in the expectations above. This means that the rank condition for identification is not satisfied and consequently the ARCH parameters may not be identifiable through this approach. An alternative may be to use a likelihood approach, which will exploit all the moments implied by the specification and the distributional assumption; this however may be particularly complicated. A convenient approximation may be to use a within group estimator on (26). This involves subtracting the individual mean of each expression on the right hand side, i.e. just replace all expressions in (26) by quantities where the individual mean has been removed. For example g_{i,a+q+1,t+q+1}g_i,a,t is replaced by . Nickell (1981) and Nerlove (1971) have shown that this estimator is inconsistent for fixed T. Effectively this implies that the estimates may be biased when T is short because the individual specific mean may not satisfy the moment conditions for short T. In practice this estimator will work well with long panel data. Meghir and Pistaferri use individuals observed for at least 16 periods. Effectively, while ARCH effects are likely to be very important for understanding behavior, there is no doubt that they are difficult to identify. A likelihood based approach, although very complex, may ultimately prove the best way forward.

Other approaches

4.1.1 Hall and Mishkin (1982)

The authors in the papers above do not write explicitly the stochastic structure of income. For example, in the statistical characterization above, permanent income is literally permanent (a fixed effect). The first paper to use micro panel data to decompose income shocks into permanent and transitory components writing an explicit stochastic income process is Hall and Mishkin (1982), who investigate whether households follow the rational expectations formulation of the permanent income hypothesis using PSID data on income and food consumption. Their setup assumes quadratic preferences (and hence looks at consumption and income changes), imposes that the marginal propensity to consume with respect to permanent shocks is 1, and leaves only the MPC with respect to transitory shocks free for estimation.

The income process is described by Eqs (3) and (4) (enriched to allow for some serial correlation of the MA type in the transitory component), so that the change in consumption is given by Eq. (5):

δC_i,a,t = Z_i,a,t + π_af_i,a,t.

Since the PSID has information only on food consumption, this equation is recast in terms of food spending (implicitly assuming separability between food and other non-durable goods):

where a is the proportion of income spent of food, and m^F is a stochastic element added to food consumption (measurement error), not correlated with the random elements of income (ζ_i,a,t and ε_i,a,t). The model is estimated using maximum likelihood assuming that all the random elements are normally distributed.

Hall and Mishkin (1982) also allow for the possibility that the consumer has some “advance information” (relative to the econometrician) about the income process.⁴⁴ Calling ϒ the degree of advance information, they rewrite their model as:

(28)

Their estimates of (28) only partly confirm the PIH. Their estimate of ϒ is 0.25 and their estimate of π (which they assume to be constant over the life cycle) is 0.29, too high to be consistent with plausible interest rates. They reconcile this result with the possibility of excess sensitivity. They note that, contrary to the theory’s prediction, cov (ΔC_a, ΔY_a−1) = 0. Hall and Mishkin suggest a set up where a fraction μ of the households overreact to changes in transitory income rather than follow the permanent income. Estimating this model, the authors find that approximately 20 percent of consumers do not follow the permanent income hypothesis.⁴⁵

4.2.1 Is the increase in income inequality permanent or transitory?

Blundell and Preston (1998) use the link between the income process and consumption inequality to understand the nature and causes of the increase in inequality of consumption and the relative importance of changes in the variance of transitory and permanent shocks. Their motivation is that for the UK they have only repeated cross-section data, and the variances of income shocks are changing over time due to, for example, rising inequality. Hence for a given cohort, say, and even ignoring measurement error, one has:

where j = 0 corresponds to the age of entry of this cohort in the labor market. With repeated cross-sections one can write the change in the variance of income for a given cohort as

Δvar(y_i,a,t) = var(Z_i,a,t) + Δvar(ε_i,a,t).

Hence, a rise in inequality (the left-hand side of this equation) may be due to a rise in “volatility” Δvar(ε_i,a,t) > 0 or the presence of a persistent income shock, var(ζ_i,a,t). In repeated cross-sections the problem of distinguishing between the two sources is unsolvable if one focuses just on income data. Suppose instead one has access to repeated cross-section data on consumption (which, conveniently, may or may not come from the same data set—the use of multiple data sets is possible as long as samples are drawn randomly from the same underlying population). Then we can see that the change in consumption inequality for a given cohort is:

assuming one can approximate the variance of the change by the change of the variances (see Deaton and Paxson, 1994, for a discussion of the conditions under which this approximation is acceptable). Here one can see that the growth in consumption inequality is dominated by the permanent component (for small r the second term on the right hand side vanishes). Indeed, assuming r ≈ 0, we can see that the change in consumption inequality identifies the variance of the permanent component and that the difference between the change in income inequality and the change in consumption inequality identifies the change in the variance of the transitory shock.⁴⁷ However, the possibility of partial insurance, serially correlated shocks, measurement error, or lack of cross-sectional orthogonality may generate underidentification.

Related to Blundell and Preston (1998) is a paper by Hryshko (2008). He estimates jointly a consumption function (based on the CRRA specification) and an income process. Based on the evidence from Hryshko (2009) and the literature, as well as the need to match the increasing inequality of consumption over the life cycle, he assumes that the income process is the sum of a random walk and a transitory shock. However, he also allows the structural shocks (i.e. the transitory shock and the innovation to the permanent component) to be correlated. In simulations he shows that such a correlation can be very important for interpreting life cycle consumption. This additional feature cannot be identified without its implications for consumption and thus provides an excellent example of the joint identifying power of the two processes (income and consumption). He then estimates jointly the income and consumption process using simulated methods of moment. In addition, just like Blundell et al. (2008b) he estimates the proportion of the permanent and the transitory shock that are insured, finding that 37% of permanent shocks are insured via channels other than savings; transitory shocks are only insured via savings.

4.2.2 Identifying an information set

Now we discuss three examples where the idea of jointly using consumption and income data has been used to identify an individual’s information set.

Cunha et al. (2005) The authors estimate what components of measured lifetime income variability are due to uncertainty realized after their college decision time, and what components are due to heterogeneity (known at the time the decision is made). The identification strategy depends on the specification of preferences and on the assumptions made about the structure of markets. In their paper markets are complete. The goal is to identify the distributions of predictable heterogeneity and uncertainty separately. The authors find that about half of the variance of unobservable components in the returns to schooling are known and acted on by the agents when making schooling choices. The framework of their paper has been extended in Cunha and Heckman (2007), where the authors show that a large fraction of the increase in inequality in recent years is due to the increase in the variance of the unforecastable components. In particular, they estimate the fraction of future earnings that is forecastable and how this fraction has changed over time using college decision choices. For less skilled workers, roughly 60% of the increase in wage variability is due to uncertainty. For more skilled workers, only 8% of the increase in wage variability is due to uncertainty.

The following simplified example demonstrates their identification strategy in the context of consumption choices. Suppose as usual that preferences are quadratic, β (1 + r) = 1, initial assets are zero, the horizon is infinite, but the consumer receives income only in two periods, t and t + 1. Consumption is therefore

Write income in t + 1 as

where is observed by both the individual and the econometrician, is potentially observed only by the individual, and is unobserved by both. The idea is that one can form the following “deviation” variables

If , there is evidence of “superior information”, i.e., the consumer used more than just to decide how much to consume in period t.

Primiceri and van Rens (2009) Primiceri and van Rens (2009) assume that consumers are unable to smooth permanent shocks, and that any attenuated response measures the amount of advance information that they have about developments in their (permanent) income. Using CEX data, they find that all of the increase in income inequality over the 1980–2000 period can be attributed to an increase in the variance of permanent shocks but that most of the permanent income shocks are anticipated by individuals; hence consumption inequality remains flat even though income inequality increases. While their results challenge the common view that permanent shocks were important only in the early 1980s (see Card and Di Nardo, 2002; Moffitt and Gottschalk, 1994), they could be explained by the poor quality of income data in the CEX (see Heathcote, 2009).

The authors decompose idiosyncratic changes in income into predictable and unpredictable permanent income shocks and to transitory shocks. They estimate the contribution of each element to total income inequality using CEX data. The log income process is specified as follows

(31)

(32)

where and ζ_i,a,t are unpredictable to the individual and ζ_i,a,t is predictable to the individual but unobservable to the econometrician. Using CRRA utility with incomplete markets (there is only a risk free bond) log consumption can be shown to follow (approximately):

(33)

From Eqs (31)–(33), the following cohort-specific moment conditions are implied:

where var(.) and cov(.) denote cross-sectional variances and covariances, respectively. Using these moment conditions, it is possible to (over)identify and for t = 1, ..., T and var(s_t) for t = 0, ..., T. The authors estimate the model using a Bayesian likelihood based approach evaluating the posterior using the MCMC algorithm. They find that predictable permanent income shocks are the main source of income inequality.

The model above cannot distinguish between predictable permanent shocks and risk sharing. To address this issue, the authors argue that if consumption does not respond to income shocks because of risk sharing, we would expect part of that risk sharing to happen through taxes and transfers and part through markets for financial assets. They show that re-estimating the model for income before taxes, income before taxes excluding financial income and for earned income before tax and transfers yields very close estimates to the baseline model (see Heathcote, 2009, for a discussion of their testing strategy).

Guvenen (2009) and Guvenen and Smith (2009) In Guvenen’s (2007) model, income data are generated by the heterogeneous income profile specification. However, individuals do not know the parameters of their own profile. In particular, they ignore the slope of life cycle profile fi and the value of the persistent component. They need to learn about these parameters using Bayesian updating by observing successive income realizations, which are noisy because of the mean reverting transitory shock. He shows that this model can be made to fit the consumption data very well (both in terms of levels and variance over the life cycle) and in some ways better than the process that includes a unit root. By introducing learning, Guveven relaxes the restriction linking the income process to consumption and as a result weakens the identifying information implied by this link. This allows the income process to be stationary and consumption to behave as if income is not stationary. Thus, from a welfare point of view the individual is facing essentially as much uncertainty as they would under the random walk model, which is why the model can fit the increasing inequality over the life cycle. In Guvenen’s model it is just the interpretation of the nature of uncertainty that has changed. The fact that the income process conditional on the individual is basically deterministic (except for the small transitory shock) has lost its key welfare implications. Thus whether the income is highly uncertain or deterministic becomes irrelevant for issues that have to do with insurance and precautionary savings: individuals perceive it as highly uncertain and this is all that matters.⁴⁸

While Guvenen (2007) calibrates the consumption profile, Guvenen and Smith (2009) use consumption data jointly with income data to estimate the structural parameters of the model. They extend the consumption imputation procedure of Blundell et al. (2008b) to create a panel of income and consumption data in the PSID. As in Guvenen (2007), they assume that the income process is the sum of a random trend that consumers must learn about in Bayesian fashion, an AR(1) process with AR coefficient below 1, and a serially uncorrelated component.

The authors estimate the structural parameters of their model by applying an indirect inference approach—a simulation based approach suitable for models in which it is very difficult to specify the criterion function.⁴⁹ The authors define an auxiliary model in which consumption and income depend on lags and leads of consumption and income, as well as growth rates of income at various lags and leads. For their estimation, the authors construct the panel of imputed household consumption by combining data from the PSID and CEX. As in Guvenen (2009) the authors find that income shocks are less persistent in the HIP case (ρ = 0.76) than in the RIP case (ρ close to one), and that there is a significant evidence for heterogeneity in income growth. In addition, they find that prior uncertainty is quite small (Λ = 0.19, meaning that about 80 percent of the uncertainty about the random trend component is resolved in the first period of life). They therefore argue that the amount of uninsurable lifetime income risk that households perceive is smaller than what is typically assumed in calibrated macroeconomic models. Statistically speaking, the estimate is very imprecise and one could conclude that everything about the random trend term is known early on in the life cycle.

4.4.1 Blundell et al. (2008b)

The consumption model considered in Blundell et al. (2008b) is given by Eq. (12), while their income process is given by (10) and (11). In their study they create panel data on a comprehensive consumption measure for the PSID using an imputation procedure based on food demand estimates from the CEX. Table 3 reproduces their main results. They find that consumption is nearly insensitive to transitory shocks (the estimated coefficient is around 5 percent, but higher among poor households), while their estimate of the response of consumption to permanent shocks is significantly lower than 1 (around 0.65, but higher for poor or less educated households), suggesting that households are able to insure at least part of the permanent shocks.

These results show (a) that the estimates of the insurance coefficients in the baseline case are statistically consistent with the values predicted by the calibrated Kaplan-Violante model of Section 2.2.2; (b) that younger cohorts have harder time smoothing their shocks, presumably because of the lack of sufficient wealth; (c) groups with actual or presumed low wealth are not able to insure permanent shocks (as expected from the model) and even have difficulties smoothing transitory shocks (credit markets can be unavailable for people with little or no collateral).

While the setting of Blundell et al. (2008b) cannot be used to distinguish between insurance and information, their paper provides a test of their assumption about richness of the information set. In particular, they follow Cunha et al. (2005) and test whether unexpected consumption growth (defined as the residual of a regression of consumption growth on observable household characteristics) is correlated with future income changes (defined also as the residual of a regression of income growth on observable household characteristics). If this was the case, then consumption contains more information than used by the econometrician. Their test of superior information reported in Table 4 shows that consumption is not correlated with future income changes.

Blundell et al. (2008b) find little evidence of anticipation. This suggests the persistent labor income shocks that were experienced in the 1980s were not anticipated. These were largely changes in the returns to skills, shifts in government transfers and the shift of risk from firms to workers.

Finally, the results of Blundell et al. (2008b) can be used to understand why consumption inequality in the US has grown less than income inequality during the past two decades. Their findings suggest that the widening gap between consumption and income inequality is due to the change in the durability of income shocks. In particular, a growth in the variance of permanent shocks in the early eighties was replaced by a continued growth in the variance of transitory income shocks in the late eighties. Since they find little evidence that the degree of insurance with respect to shocks of different durability changes over this period, it is the relative increase in the variability of more insurable shocks rather than greater insurance opportunities that explains the disjuncture between income and consumption inequality.

4.4.2 Solution 1: the quasi-experimental approach

The approach we discuss in this section does not require estimation of an income process, or even observing the individual shocks.⁵⁴ Rather, it compares households that are exposed to shocks with households that are not (or the same households before and after the shock), and assumes that the difference in consumption arises from the realization of the shocks. The idea here is to identify episodes in which changes in income are unanticipated, easy to characterize (i.e., persistent or transient), and (possibly) large.

The first of such attempts dates back to a study by Bodkin (1959), who laid down fifty years ago all the ingredients of the quasi-experimental approach.⁵⁵ In this pioneering study, the experiment consists of looking at the consumption behavior of WWII veterans after the receipt of unexpected dividend payments from the National Service Life Insurance. Bodkin assumes that the dividend payments are unanticipated and represent a windfall source of income, and finds a point estimate of the marginal propensity to consume non-durables out of this windfall income is as high as 0.72, a strong violation of the permanent income model.⁵⁶

The subsequent literature has looked at the economic consequences of illness (Gertler and Gruber, 2002), disability (Stephens, 2001; Meyer and Mok, 2006), unemployment (Gruber, 1997; Browning and Crossley, 2001), and, in the context of developing countries, weather shocks (Wolpin, 1982; Paxson, 1992) and crop losses (Cameron and Worswick, 2003). Some of these shocks are transitory (i.e. temporary job loss), and others are permanent (disability); some are positive (dividend pay-outs), others negative (illness). The framework in Section 2 suggests that it is important to distinguish between the effects of these various types of shocks because, according to the theory, consumption should change almost one-for-one in response to permanent shocks (positive or negative), but may react asymmetrically ifshocks are transitory. Indeed, ifhouseholds are credit constrained (can save but not borrow) they will cut consumption strongly when hit by a negative transitory shock, but will not react much to a positive one.

Recent papers in the quasi-experimental framework look at the effect of unemployment shocks on consumption, and the smoothing benefits provided by unemployment insurance (UI) schemes. As pointed out by Browning and Crossley (2001) unemployment insurance provides two benefits to consumers. First, it provides “consumption smoothing benefits” for consumers who are liquidity constrained. In the absence of credit constraints, individuals who faced a negative transitory shock such as unemployment would borrow to smooth their consumption. If they are unable to borrow they would need to adjust their consumption downward considerably. Unemployment insurance provides some liquidity and hence it has positive welfare effects. Second, unemployment insurance reduces the conditional variance of consumption growth and hence the need to accumulate precautionary savings.

One of the earlier attempts to estimate the welfare effects of unemployment insurance is Gruber (1997). Using the PSID, he constructs a sample of workers who lose their job between period t − 1 and period t, and regresses the change in food spending over the same time span against the UI replacement rate an individual is eligible for (i.e., potential benefits). Gruber finds a large smoothing effect of UI, in particular that a 10 percentage point rise in the replacement rate reduces the fall in consumption upon unemployment by about 3 percent. He also finds that the fall in consumption at zero replacement rates is about 20 percent, suggesting that consumers face liquidity constraints.

Browning and Crossley (2001) extend Gruber’s idea to a different country (Canada instead of the US), using a more comprehensive measure of consumption (instead of just food) and legislated changes in UI (instead of state-time variation). Moreover, their data are rich enough to allow them to identify presumably liquidity constrained households (in particular, their data set provide information on assets at the time of job loss). Browning and Crossley estimate a small elasticity of expenditures with respect to UI benefit (5 percent). But this small effect masks substantial heterogeneity, with low-assets households at time of job loss exhibiting elasticities as high as 20 percent. This is consistent with the presence of liquidity constraints.

A critique of this approach is that the response of consumption to unemployment shocks is confounded by three sets of issues (similar arguments apply to papers that look at unpredictable income changes due to illness or disability, as in Stephens, 2001). First, some of these shocks may not come as a surprise, and individuals may have saved in their anticipation. For example, being laid off by Chrysler in 2009 should hardly come as a surprise. Ideally, one would overcome this problem by, say, matching job accident data or firm closure data with consumption data. Second, the theory predicts that consumers smooth marginal utility, not consumption per se. If an unemployment shock brings more leisure and if consumption is a substitute for leisure, an excess response of consumption to the transitory shock induced by losing one’s job does not necessarily represent a violation of the theory. Finally, even ifunemployment shocks are truly fully unanticipated, they may be partially insured through government programs such as unemployment insurance (and disability insurance in case of disability shocks). An attenuated consumption response to a permanent income shock due to disability may be explained by the availability of government-provided insurance, rather than representing a failure of the theory. Therefore a complete analysis of the impact of unemployment or disability shocks requires explicit modeling of the type of insurance available to individuals as well as of the possible interactions between public and private insurance.

The above discussion suggests that it might be easier to test the theory in contexts in which insurance over and above self-insurance is not available, such as in developing countries. Gertler and Gruber (2002) look at the effect of income shocks arising from major illness on consumption in Indonesia. They find that while people are able to smooth the effect of minor illnesses (which could be interpreted as transitory shocks, or anticipated events), they experience considerably more difficulty in smoothing the impact of major illnesses (which could be interpreted as permanent shocks).

Wolpin (1982) and Paxson (1992) study the effect of weather shocks in India and Thailand, respectively. In agricultural economies, weather shocks affect income directly through the production function and deviations from normal weather conditions are truly unanticipated events. Wolpin (1982) uses Indian regional time series data on rainfall to construct long run moments as instruments for current income (which is assumed to measure permanent income with error). The estimated permanent income elasticity ranges from 0.91 to 1.02 depending on the measure of consumption, thus supporting strongly the permanent income model. Paxson (1992) uses regional Thai data on weather to measure transitory shocks and finds that Thai consumers have a high propensity to save out of transitory weather shocks, in support of the theory. However, she also finds that they have a propensity to save out of permanent shocks above zero, which rejects a strong version of the permanent income hypothesis.

Studies using quasi-experimental variation to identify shocks to household income have the obvious advantage that the identification strategy is clear and easy to explain and understand. However, these studies’ obvious limitation is that they capture only one type of shock at a time, for instance illness, job loss, rainfall, extreme temperatures, or crop loss. One may wonder, for example, whether the Gruber (1997) and Browning and Crossley (2001) estimates obtained in a sample of job losers have external validity for examining the effect of other types of shocks (especially those that are much harder to insure, such as shocks to one’s productivity).

A second limitation of the approach is that some of the income shocks (in particular, unemployment and disability shocks), cannot be considered as truly exogenous events. For instance, for some people unemployment is a voluntary choice, and for others disability could be reported just to obtain benefits (a moral hazard issue). For this reason, not all income variability is necessarily unanticipated, or exogenous to the agent (Low et al., forthcoming). The lesson of the literature is that identifying episodes of genuine exogenous and unanticipated income changes is very difficult. One such case is weather conditions, to the extent at least to which people don’t move to different regions to offset bad weather conditions.

4.4.3 Solution 2: subjective expectations

As pointed out in Sections 4.1 and 4.2, identifying income shocks is difficult because people may have information that is not observed by the econometrician. For instance, they may know in advance that they will face a temporary change in their income (such as a seasonal lay-off). When the news is realized, the econometrician will measure as a shock what is in fact an expected event. The literature based on subjective expectations attempts to circumvent the problem by asking people to report quantitative information on their expectations, an approach forcefully endorsed by Manski (2004). This literature relies therefore on survey questions, rather than retrospective data (as in Section 4.2), to elicit information on the conditional distribution of future income, and measures shocks as deviations of actual realizations from elicited expectations.

Hayashi (1985) is the first study to adopt this approach. He uses a four-quarter panel of Japanese households containing respondents’ expectations about expenditure and income in the following quarter. Hayashi works with disaggregate consumers’ expenditure, allowing each component to have a different degree of durability. He specifies a consumption rule and, allowing for measurement error in expenditures, estimates the covariances between expected and unexpected changes in consumption and expected and unexpected changes in income. His results are in line with Hall and Mishkin (1982), suggesting a relatively high sensitivity of consumption to income shocks.

Pistaferri (2001) combines income realizations and quantitative subjective income expectations contained in the 1989–93 Italian Survey of Household Income and Wealth (SHIW) to point identify separately the transitory and the permanent income shocks. To see how subjective income expectations allow the estimation of transitory and income shocks for each household, consider the income process of Eqs (3) and (4). Define E(X_i,a,t|Ω_{i,a−1,t−1}) as the subjective expectation of (X_i,a,t given the individual’s information set at age a − 1. It is worth pointing out that Ω_{i,a−1,t−1} is the set of information possessed at the individual level; the econometrician’s information set is generally less rich. The assumption of rational expectations implies that the transitory shock at time t can be point identified by:

(36)

Using Eqs (3), (4) and (36), the permanent shock at time t is identified by the expression:

i. e., the income innovation at age a adjusted by a factor that takes into account the arrival of new information concerning the change in income between a and a + 1. Thus, the transitory and permanent shocks can be identified if one observes, for at least two consecutive time periods, the conditional expectation and the realization of income, a requirement satisfied by the 1989–93 SHIW Pistaferri estimates the saving for a rainy day equation of Campbell (1987) and finds that consumers save most of the transitory shocks and very little of the permanent shocks, supporting the saving for a rainy day model.

Kaufmann and Pistaferri (2009) use the same Italian survey used by Pistaferri (2001), but different years (1995–2001) to distinguish the superior information issue from the insurance issue mentioned in Section 4.2. Their empirical strategy is to consider the covariance restrictions implied by the theory on the joint behavior of consumption, income realizations, and subjective quantitative income expectations.

Their results are reproduced in Table 5. Their most general model separates transitory changes in log income into anticipated (with variance ), unanticipated , and measurement error ; separates permanent changes in income in anticipated and unanticipated allows for measurement error in consumption and subjective income expectations ( and , respectively), and allows for partial insurance with respect to transitory shocks (Ψ) and permanent shocks (Φ).

In column (1) they put themselves in the shoes of a researcher with access to just income data. This researcher cannot separate anticipated from unanticipated changes in income or transitory changes from measurement error, so she assumes that measurement error is absent and all changes are unforecastable, resulting in upward biased estimates of , and ,. In column (2) they add consumption data. The researcher is still unable to separate anticipated from unanticipated, so any “superior information” is loaded onto the insurance coefficients Ψ and Φ. In particular, the data provide evidence of some insurance with respect to permanent and transitory shocks. Note that unlike what is predicted by the traditional version of the PIH, the transitory shock is not fully insured, perhaps because of binding borrowing constraints (see Jappelli and Pistaferri (2006)). In column (3) one adds data on subjective income expectations and the model is now overidentified. A number of interesting facts emerge. First, the transitory variation in income is split between the anticipated component (about 50%), the unanticipated component (20%) and measurement error (30%). This lowers the estimated degree of insurance with respect to transitory shocks. Similarly, a good fraction of the permanent variation (about 1/3) appears anticipated, and this now pushes the estimated insurance coefficient towards 1−i.e., these results show evidence that there is no insurance whatsoever with respect to permanent shocks.

There are a few notes of caution to add to the commentary on these results. First, the overidentifying restrictions are rejected. Second, while the economic significance of the results is in accordance with the idea that part of the estimated smoothing effects reflect information, the standard errors are high, preventing reliable inference.