Abstract
I empirically investigate precautionary savings under liquidity constraints in Italy using a unique indicator of subjective variance of income growth to measure the strength of the precautionary motive for saving, and a variety of surveybased indicators of liquidity constraints. The main contribution of the paper is twofold. First of all, I attempt to differentiate between the standard precautionary saving caused by uncertainty from the one due to liquidity constraints using an endogenous switching regression approach, which allows me to cope with endogeneity issues associated with sample splitting techniques. Second, I move one step further with respect to previous studies on consumption behaviour by taking explicitly expected liquidity constraints into account. I eventually found the precautionary motive for savings to be stronger for those households who face binding constraints, or expect constraints to be binding in the future. Indeed, a complementarity relation exists between precautionary savings and liquidity constraints.
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Notes
 1.
According to the life cycle permanent income hypothesis (LCPIH), changes in consumption should be unrelated with anticipated changes in income and other variables in consumers’ information set. Under the certainty equivalence (CE) restrictive assumptions (perfect capital markets, separable and additive preferences, equality between interest rate and the subjective discount factor and quadratic utility) it is possible to obtain a closed form solution for consumption.
 2.
When risk is taken into consideration in the optimisation problem, the prudence coefficient, given by the ratio between the third and the second derivative of the lifetime utility function, represents the relevance of the precautionary motive for savings (Kimball 1990). Allowing a utility function specification with a non zero third derivative is, therefore, the key requirement to take into consideration the effect of uncertainty on future consumption and wealth allocation.
 3.
In this regard, see Ludvigson and Paxson (1997).
 4.
In this regard see Dynan (1993), who estimate a constant relative risk aversion utility function (CRRA) using consumption variability as a measure of future risk and Banks et al. (2001), who rely on Skinner (1988) approximation, and estimate the Euler equation using the variance of income innovations to measure the strength of precautionary savings.
 5.
In this regard, using patterns of variation across age, education, industry and occupation, might not be helpful. A self selection problem may arise, since people who are more risk tolerant may both choose to work in a relatively risky industry and not save much. Therefore, the effect of an exogenous change in risk may lead to a downward bias in the estimation of the coefficient associated to the measure of uncertainty.
 6.
 7.
In this regard, Carroll and Kimball (2001) state that expected liquidity constraints might affect households’ consumption choices as well as effective ones. Moreover, Jappelli and Pistaferri (2000) argue that prudent consumers may save in anticipation of future constraints, thus do not showing excess sensitivity to future income.
 8.
Since here we consider real variables, we need income and inflation expectations. Because of missing observations about inflation expectations, the sample reduces when we consider the first and second moment of real income growth
 9.
See the appendix for the derivation of the first and second moment of real income growth.
 10.
 11.
Only if utility is exponential and income is a random walk there is a one to one correspondence between income risk and consumption risk in the Euler equation. Otherwise, the relation between the two is nonlinear, and it depends on the specification of the utility function and on the income process (Jappelli and Pistaferri 2000).
 12.
The significance of the coefficient associated to the variance cannot be due to measurement errors. In an OLS context measurement error in an independent variable tends to bias the coefficients towards zero. In this regard, measurement error cannot explain, alone, a significant coefficient of income risk. For the same reason, measurement error in expected income may be the cause of the bias (towards zero) of the excess sensitivity coefficient.
 13.
In this regard, Kennikell and Lusardi (2004) using a subjective measure of desired precautionary saving notice that other risks beyond income risk are responsible for households’ level of precautionary saving. Regarding health risk, Nocetti and Smith (2010) evaluate how uncertainty affects the demand of curative and preventive care.
 14.
In this regard, see Guiso and Jappelli (2002).
 15.
All variables in the right and in the left hand side are expressed in real terms using the price index of consumer prices (ISTAT 2009).
 16.
If \(\text{ T}\rightarrow \infty \) , the forecast error goes to 0. However, in panels with small \(\text{ T}\) and big \(\text{ N}\) there is no guarantee that the forecast error goes to 0 as \(\text{ N}\rightarrow \infty .\) As specified by Jappelli and Pistaferri, excess sensitivity may arise spuriously from the misspecification of the stochastic structure of the forecast error. However, as the authors point out, subjective expectations help to mitigate the problems one faces when testing for excess sensitivity with short panels. Indeed, they provide a guide to model the stochastic structure of the forecast error, arguing that forecast errors in income can be used to extract useful information about the structure of forecast errors in consumption, which is unobservable. Moreover, they eventually found forecast errors in consumption to be correlated not only with time dummies, but also with education, region and occupation. Therefore, the forecast error is modelled as to contain an aggregate component which is unevenly distributed across population groups and an idiosyncratic component that averages out in the cross section. Moreover, group dummies (education, region, employment status) are used as instruments in order to predict income growth. The problem tackled by Jappelli and Pistaferri is not central to my analysis. Indeed, the fact that the forecast error in consumption does not go to zero in short panels affects the coefficients associated to income growth and, therefore, the test for excess sensitivity. In this regard, the subjective expectation about income growth is used by Jappelli and Pistaferri (2000) as an instrument to tackle endogeneity of the effective income growth. However, this issue does not apply to the variance, which is unlikely to be endogenous, thus leading to spurious evidence in favour of precautionary saving.
 17.
One of the objection that may be done to this specification refers to the fact that consumption growth is over 2 years (biannual survey), i.e. \(\text{ c}_{\mathrm{i},1991}\text{c}_{\mathrm{i},1989}\), whereas income expectations are for the twelve months following the interview, i.e. \(\text{ E}_{1989}(\text{ y}_{\mathrm{i,May}1991}\text{y}_{\mathrm{i,May}1990})\) (interviews are carried out in May). Hence, consumption growth and income growth expectations are not directly comparable. However, we are not directly interested in testing for excess sensitivity, so that these problems are not directly relevant for our estimates.
 18.
I do not introduce labour supply indicators in my estimation. My aim is indeed to focus on the coefficient associated to the variance of income growth, looking at the strength of precautionary motive for saving, instead of testing for excess sensitivity. It may be that the variance of income growth is correlated with labour supply. However, even following Jappelli and Pistaferri (2000) the results of the IV estimation do not affect the significance of the coefficient associated to the measure of uncertainty.
 19.
Lee and Sawada (2010) perform three different specifications, using several sets of instruments in order to correct for the endogeneity of the uncertainty term.
 20.
Indeed, Jappelli and Pistaferri (2000) got a coefficient equal to 5.67.
 21.
See Guiso and Paiella (2008) for an explanation of the method used to calculate the coefficient of absolute risk aversion.
 22.
See the appendix as for frequency of households which can be considered as constrained/unconstrained according to several classification criteria.
 23.
Another objection related to the indicator of liquidity constraints is that there may be liquidity constrained households that simply did not ask for a loan in the specific year of the survey. However, here we take into consideration liquidity constraints in a broader sense. Indeed, a household can be considered as constrained when their desired level of consumption is higher than the effective one (Lee and Sawada 2010). The fact that our indicator of constraints includes discouraged households (who may have wished to ask for a loan, but they were prevented to do that) partly solves this problem. In the last paragraph we disentangle demand side from supply side effects.
 24.
Earlier approaches only included in \(Z\) financial variables. Zeldes (1989), for example, use a simple asset/income ratio in order to determine who is liquidity constrained. However, as Jappelli et al. (1998) mention, this approach might lead to misclassifications. Indeed, wealth is a good indicator of liquidity constraints only if there is a roughly monotonic relation between the two. Second, sample splits based on wealth are bound to be highly imperfect because assets and asset income are often poorly measured. Finally, households might have a high level of wealth, but some assets may be committed to pay for a mortgage for example, so they are not available to smooth nondurable consumption.
 25.
The inclusion of subjective moments of income growth as explanatory variables of the probability of being constrained can be explained by taking into consideration the fact that the measure of constraints we are using encompasses both demand and supplyside factors. Subjective average and variability of future income growth—although not affecting the probability the loan is rejected—are likely to affect household probability of asking for a loan. This is true, in particular, when expected constraints are taken into account.
 26.
This method gives more efficient estimates rather than Heckman twostep approach.
 27.
Data comes from the Bank of Italy ‘Base Informativa Pubblica’.
 28.
Expected inflation should affect wealth at \((t+1)\) only if wealth is invested in some assets. Indeed, one should have detailed information on the amount invested in each asset by households, as well as on the return of each asset. While the former information is available in the SHIW, the latter is difficult to gather. Since a great part of households’ wealth is invested in the house, whose value is affected by expected inflation, I assume wealth augmented by expected inflation to be a good proxy of expected wealth at \(t+1\).
 29.
Note that the dependent variable in the probit regression is a dummy which takes value 1 if liquidity constraint indicators lb2 or lb3 take value 1, 0 otherwise. We tried alternative specifications using lb2 or lb3 alone, or lb1. Results remain basically unaffected.
 30.
As shown in the appendix, the average value of the estimated probability is around 13 %.
 31.
A switching regression has also been performed. Empirical results are basically the same; however, correlation coefficient rho1 and rho2 are not significant. Results are not reported but they are available upon request.
 32.
0.6 is the closest value to the average of the estimated probability. Moreover, this is the value which gives the highest value of maxlikelihood. This is in line with models of endogenous switching with unknown sample selection rule (see Hotchkiss 1991).
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Acknowledgments
I am especially grateful to Francesco Nucci for his patient and precious supervising activity. I also thank Monica Paiella, two anonymous referee and seminar participants at the Tor Vergata PhD students conference for helpful suggestions. Of course, I take full responsibility for any errors or omissions. I acknowledge financial support from Regione Autonoma della Sardegna, ‘Master and Back’ research grant.
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Appendix
Appendix
List of variables used in the empirical analysis:

DC log rate of growth of nondurable consumption

Varredn variance of nominal income rate of growth

Eredn expected rate of growth of nominal income

Varredr variance of real income rate of growth

Eredr expected rate of growth of real income

Male dummy \( = \) 1 if the household head is male, 0 otherwise

Age age of the household head

Dncomp change in family size

Farmer dummy \( = \) 1 if the household head is a farmer

Entrepreneur dummy \( = \) 1 if the household head is an entrepreneur

\(W \)household net wealth (financial asset + real asset –financial liabilities)

\(W2\) squared household net wealth

Wrisk coefficient of risk aversion (as in Guiso and Paiella 2008)*W

acom2 dummy \( = \) 1 if number of inhabitants is between 20,000 and 40,000

acom3 dummy \( = \) 1 if number of inhabitants is between 40,000 and 500,000

acom4 dummy \( = \) 1 if number of inhabitants is \(>\) 500,000

South dummy \( = \) 1 if the household head lives in the south

Bank counters number of bank counters per person (regional basis)
Sample selection
The panel component of the SHIW includes 3,629 observations. I only considered household heads. I dropped all the observations that do not appear for at least 2 years. Moreover, I dropped all the observations with inconsistent data and those lacking data on subjective expectations.
Variables definition
Inflation uncertainty
On this table we have indicated some classes of inflation. We are interested in knowing your opinion about inflation twelve months from now. Suppose now that you have 100 points to be distributed between these intervals. Are there intervals which you definitely exclude? Assign zero points to this intervals. How many points do you assign to each of the remaining intervals? For this and the following variable the intervals of the table shown to the person interviewed are the same. The intervals are: \(>\)25, 20–25, 15–20, 13–15, 10–13, 8–10, 7–8, 6–7, 5–6, 3–5, 0–3, \(<\)0 percent. In case it is less than zero, the person is asked: How much less than zero? How many points would you assign to this class?
Earnings uncertainty
We are interested in knowing your opinion about labour earnings or pensions twelve months from now. Suppose now that you have 100 points to be distributed between these intervals. Are there intervals which you definitely exclude? Assign zero points to this intervals. How many points do you assign to each of the remaining intervals?
Derivation of subjective moments of real income growth
Following Guiso et al. (1992), define \(z\) as the percentage growth rate of nominal earnings, \(\pi \) as the rate of inflation and \(x\) as the rate of growth of real earnings, where: \(z=x+\pi ,\) and
\(\sigma _x\) can, therefore, be rewritten as:
Since the variance of nominal income can be either positive or zero, there are 4 possible cases:
Liquidity constraints
1989

(1)
In 1989 did your household apply to a bank or a financial company for a loan or a mortgage? Yes No

(2)
Was the application granted in full, in part or rejected? Granted in full Granted in part Rejected

(3)
Why didn’t you apply for a loan in 1989? no need for a loan I thought the application would be rejected
1991

(1)
In 1991 has your application for a loan been rejected or granted in part? Yes no
In 1991 did you or another member of your household consider the possibility of applying to a bank or a financial company for a loan or a mortgage but then change his/her mind thinking that the application would be rejected?
Table 8 reports descriptive statistics regarding different definitions of liquidity constraints. According to the first definition \((b1)\) households are constrained if they ask for a loan but their request is rejected, or if they are ‘discouraged’ (they do not ask for a loan because they think they will be rejected). In order to increase the number of observations in the subsamples, I consider other indicators of liquidity constraints. According to the second definition \((b2)\), households are constrained if their total net wealth is lower than 2 months’ income (Hayashi 1985b). The third definition \((b3)\) uses households’ past level of indebtness in order to distinguish between constrained and unconstrained households. According to the latter definition, an household is considered as constrained if the ratio financial liabilities/(real \( + \) financial asset) is greater than 0.5.
Table 8 reports descriptive statistics regarding estimated probability of future (expected) liquidity constraints. e_prob is obtained in two step. First, the probability of liquidity constraint at time \(t\) is estimated, using a joint indicator (lb2 and lb3) as dependent variable. Then, coefficients obtained in the first step are used to estimate households’ expected probability of being liquidity constrained in the future. In order to get credible values of explanatory values at time \(t+1\). An household is defined as constrained if e_prob is higher than 0.6, unconstrained otherwise.
cond_prob is obtained with the same twostep procedure used to obtain .e_prob. However, the probability estimated in the first step is obtained by a conditional probit. A household is defined as constrained if cond_prob is higher than 0.6, it is taken as unconstrained otherwise.
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Deidda, M. Precautionary saving under liquidity constraints: evidence from Italy. Empir Econ 46, 329–360 (2014). https://doi.org/10.1007/s001810120677y
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Keywords
 Consumption
 Precautionary saving
 Liquidity constraints
 Switching regression
JEL Classification
 E21
 C21