How do House Prices Affect Household Savings?
Empirical Evidence from a Dynamic Model using a
Dutch Administrative Panel Dataset
By
Yoram Vanmaekelbergh1
10003416
Supervisor Prof. dr. J. F. Kiviet
Master Thesis Econometrics August, 2015
Abstract
This study investigates the effect of a house price shock on savings behaviour of Dutch households. Using a random subsample of a unique administrative dataset, it is established that models of previous research within this field do not appropriately take into account the dynamics within the relationship under study. It is argued that these dynamics are satisfactorily modelled in an autoregressive distributed lag model estimated by the Arellano Bond estimator. Results from a model which is sound according to all dynamic panel data misspecification tests elicit that a house price increase is associated with a long term savings elasticity ranging between 0.21 and 0.60, albeit with a 95% confidence interval as large as [-0.59, 1.79]. Comparing the long term house price elasticities with respect to savings among different municipalities under a ceteris paribus condition demonstrates that the results can best be explained by a precautionary savings motive, prospects of increased future housing costs and the collateral hypothesis. However, the results appear to be relatively sensitive to the employed estimation method, used model and maintained definitions, which implies that the results should be interpreted with a sound level of caution.
JEL Classification: D11, D12, D14, E21, R21, R22.
Keywords: household savings, house prices, collateral hypothesis, wealth hypothesis, financial liberalization hypothesis, common factor hypothesis, precautionary savings motive, dynamic model, autoregressive distributed lag model, Arellano Bond estimator.
1 Yoram Vanmaekelbergh, MSc Student Econometrics, Amsterdam School of Economics,
Section Econometrics and Statistics, University of Amsterdam. I would like to thank my supervisor Jan Kiviet (Nanyang Technological University) for his supervision and feedback. I am grateful to Remco Mocking (Netherlands Bureau for Economic Policy Analysis) for his supportive feedback and for providing the unique dataset.
“ ‘Propose what can be done’, they never stop repeating to me. It is as if I were told, ‘propose doing what is done’.”
Rousseau (1762)
Statement of Originality
This document is written by Student Yoram Vanmaekelbergh who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
1 Introduction... 4
2 Literature review ... 5
3 The data ... 10
4 Methodology ... 12
4.1 Reanalyzing the model of Beers, Bijlsma and Mocking (2015) ... 12
4.2 The auto-regressive distributed lag model ... 16
4.3 Estimation method ... 18
4.4 Regression equations ... 20
5 Results ... 26
5.1 Results model (5) ... 26
5.2 Results model (6) ... 30
5.3 Inference based on the interaction terms ... 34
5.3.1 Heterogeneity in the housing wealth effect with respect to age ... 35
5.3.2 Heterogeneity in the housing wealth effect with respect to percentage declined houses ... 36
5.3.3 Heterogeneity in the housing wealth effect with respect to presence of children ... 37
5.3.4 Heterogeneity in the housing wealth effect with respect to loan to value ratio and leverage ... 38
5.3.5 Heterogeneity in the housing wealth effect with respect to disposable income ... 40
5.4 Sensitivity analysis ... 41
5.4.1 Analysis of model (6) with an extended definition of savings ... 42
5.4.2 Analysis of model (6) with additional interactions ... 42
5.4.3 Sensitivity with respect to the used orthogonality conditions ... 45
5.4.4 Analysis of model (6) with explicitly disentangling house price changes in an expected and unexpected part ... 45
5.4.5 Outlier analysis ... 46
Conclusion and discussion ... 47
Appendix A1: Variables used in model (6) ... 51
Appendix A2: Example of elasticity derivations ... 52
Appendix A3: Arellano Bond estimator ... 54
Appendix A4: The housing market in the Netherlands ... 58
Appendix A5: Disentangling house prices in expected and unexpected house price changes . 61 References ... 64
1 Introduction
The relationship between household savings and house prices has been of ongoing interest in the economic literature since the 2007 economic recession. More recently, house price drops which simultaneously occurred with decreased consumption has increased the attention of policy makers to the role housing wealth plays in the economic decisions of a household. This increased interest is displayed in the growing number of scientific papers as well as the increased number of advices of different policy organizations about this relationship.
The finding that house prices and savings move synchronized is widespread among previous research. However, the causal mechanism behind this relationship as well as the size of the effect is less clear. The current state of the literature has identified four main hypotheses by which an increased house price may lead to a decrease in household savings. Firstly, an unexpected and permanent increase in lifetime wealth due to risen house prices may directly decrease savings according to the wealth hypothesis. Alternatively, increasing house prices may lift borrowing constraints for some households, which may indirectly lead to decreased savings. Conversely, the relationship between house prices and savings may not be causal: a common factor, such as households’ expectations about their future income, may affect both simultaneously. Besides, a policy change resulting in financial liberalization may induce the pattern of decreased savings while house prices increase at the same time.
The principal objective of this paper is to disentangle the causal mechanism behind the observed correlation between house prices and savings. This is important because housing wealth is an extensively discussed policy instrument in the Netherlands, whilst each causal mechanism entails different policy implications.
The most substantive contribution of this paper to the existing literature is the dynamic model specification in which the relationship is studied. This is a huge difference from previous studies, which all use static models for inference about housing wealth effects. Another contribution to the existing literature is that the effects of aggregating the data to different levels, i.e. to household level and to municipality level are compared with each other, unlike previous research which only studies either macro-economic effects or employed a micro-economic approach. Lastly, minor contributions are summarized by the interpretation of the obtained effects in light of the Dutch housing market, applying the model to a unique administrative panel dataset containing information on the household level and accounting explicitly for heterogeneity in the long term elasticity to save out of housing wealth.
Overall, a small positive long term house price elasticity with respect to savings is found, which is a novel finding within this field. This finding emerges from a model in which all data are aggregated on municipality level, because models on household level do not seem credible enough for valid inference. By taking into account the heterogeneity among municipalities, it is concluded that the results obtained can best be explained by a combination of a
precautionary savings motive and increased housing costs prospects, since the largest long term propensities to save out of housing wealth are found for highly leveraged municipalities with young home owners. In addition, evidence for a collateral effect is found, since households with an extremely low disposable income tend to dissave if their housing wealth increases. These results are robust to several applied estimation methods and sensitivity checks, but only few results reach significance at the 5% significance level.
The remainder of this study is organized as follows. Section 2 describes the aforementioned explanations for the correlation between house prices and savings in more detail and reviews the effects established by previous studies. Section 3 describes the unique administrative dataset used in the present study. Section 4 presents why the static models used in previous research are inappropriate and argues that the autoregressive distributed lag model estimated with the Arellano Bond estimator is a superior alternative. Section 5 discusses the results emerging from this model, including a number of robustness checks. The study ends with the conclusion and the limitations.
2 Literature review
The finding that house prices and savings (or similarly consumption) move synchronized, as can be seen in figure 1, is widespread among previous research. However, the causal mechanism behind their co-movement is less clear. This section discusses the results of previous research on the aforementioned relationship based on micro data. Focal in this section are the two causal links which are found the most often: evidence for a wealth effect is compared with evidence for a collateral effect. Two studies carried out with a Dutch dataset are discussed in greater detail at the end of this section, because this study also uses Dutch data. In the last part of this section, studies taking into account possible heterogeneity of the housing wealth effect on savings are addressed, because such a heterogeneity distribution is also derived in the present study.
Figure 1
House price development in real prices (2000 is base year) in the period
2000-2014 as measured by Statistics Netherlands’ house price index together with
level of savings development in real prices (2000 is base year) as measured by
the Dutch Central Bank’s total household savings.
The first explanation for the co-movement of house prices and savings can be summarized as the wealth effect. According to this assertion, households behave as is supported by the life-cycle hypothesis or permanent income hypothesis. These hypotheses state that households adjust their spending upon receiving unexpected information about their lifetime wealth while anticipated changes in wealth are already incorporated into this spending plan (see for example Modigliani, 1966). If new information about a household’s house price becomes available, this information is assimilated in the household’s future spending plan. For example, if a household receives information that its house has increased in value and that this increase is permanent, this may result in decreased savings (or equivalently, increased consumption) by a risen perceived life-time wealth stemming from this information, ceteris paribus. Researchers argue that they find evidence for the wealth hypothesis if they find a higher propensity to consume out of housing wealth (or equivalently a lower propensity to save out of housing wealth) for older households, because those households have less time to assimilate the new information in their life time consumption plan. In addition, a reaction to house price changes which are argued to be unexpected and permanent rather than expected is also seen as evidence for the wealth hypothesis, as expected house price movements should already be incorporated in the households’ life time consumption plan.
This wealth effect is supported by a great deal of recent research. For example Campbell and Cocco (2008) find a strong effect of consumption out of housing wealth using UK micro data. In line with the life cycle hypothesis, they find that old home owners increase their consumption more than young home owners after house prices have increased. Next to that, Lehnert (2004) finds
0 20 40 60 80 100 120 140 160 180 200 20 00 20 01 20 02 20 03 20 04 20 05 20 06 20 07 20 08 20 09 20 10 20 11 20 12 20 13 20 14 Savings index House price index
evidence for a wealth effect by analyzing US micro panel data on consumption and house prices. In that study, the wealth effect is again substantiated with heterogeneity across age in the assessed marginal propensities to consume out of housing wealth. In addition, Disney, Henley and Stears (2001) find a significantly higher marginal propensity to save out of housing wealth for UK households planning to move, which they argue to correspond to a wealth effect as well. Continuing, Burrows (2013) also finds a wealth effect by analyzing the decision to withdraw equity out of housing wealth using a probit model applied to UK micro data. Among her findings is that households do withdraw equity after an unexpected house price increase, while they don’t withdraw equity after an expected increase in house prices. Lastly, the wealth effect is also supported by the combined micro and macro data approach of Clancy, Cussen and Lydon (2014) for the country of Ireland, since they find stronger increases in durable and non-durable consumption out of housing wealth for old households.
The support for a wealth effect seems large, but the support for the
collateral effect is at least as large. According to the collateral hypothesis, households regard their houses as equity with which they are able to obtain a secured loan. An increased house price may facilitate borrowing a larger amount of money, resulting in increased household consumption or decreased household savings. Moreover, increased house prices may give access to cheaper loans, because the households’ balance sheet has improved. This lower cost of finance due to decreased external finance premiums may translate in increased spending or decreased saving. Researchers state that they find evidence for the collateral effect if credit constrained households, e.g. younger households, households with a loan to value ratio greater than one or households planning to move, display stronger propensities to consume out of housing wealth.
Presence of the collateral effect is found by Gathergood (2012); he analyzes the responses to a question of an UK survey which asks whether households would decrease their consumption if house prices decreased with 10%. His results elicit that households who claim to be collateral constrained are 30% more likely to decrease their consumption. Furthermore, Atalay, Whelan and Yates (2014) find strong evidence for a collateral effect by analyzing consumption data from an Australian and Canadian panel data survey. Among their findings is a larger marginal propensity to consume out of housing wealth for middle aged households compared to that of old households, which typically corresponds to a collateral effect. In addition, Mian, Rao and Sufi (2013) find a collateral effect by analyzing a unique dataset derived from Mastercard purchases across different US counties. They argue for the collateral effect by the significantly larger marginal propensity to consume out of increased house prices for highly leveraged households. Finally, Browning, Gortz and Leth-Petersen (2013) reach similar conclusions by analyzing consumption expenditure in an administrative panel dataset of the Danish population. Their findings are in line with the collateral effect, since they find that marginal propensities to consume out of housing wealth become stronger after a reform
introducing possibilities for equity withdrawal. Moreover, Campbell and Cocco (2008), Slacelek (2009) and Lehnert (2004) find evidence for a collateral effect next to evidence for a wealth effect.
An explanation related to the collateral hypothesis is the precautionary
savings motive. In order to improve their financial situation in the future,
households may save housing wealth gains. Households with weaker balance sheets, i.e. highly leveraged households or households with a higher loan to value ratio, increase their savings even more as compared to less indebted households. Evidence for such an effect is found by Dynan (2012).
Alternative explanations for the correlation between house prices and savings also exist. The synchronized movement could for example be caused by
a common factor. This assertion states that both consumption and house prices
are simultaneously driven in the same direction by a particular factor. For example, expectations about productivity may influence some household specific factors, which may induce a common trend for house prices and consumption. Attanasio et al. (2009) find support for this explanation by analyzing a pseudo panel based on the British population. Another common factor possibly driving the co-movement between house prices and consumption could be future income and cost prospects: households may decrease their consumption (or equivalently increase their savings) after house prices have increased, because of prospects of risen future housing expenses. This explanation is partly supported by King (1990).
A final explanation can be resumed as the financial liberalization hypothesis. Financial liberalization, i.e. financial reforms which make it easier to obtain a loan, may result in increased house prices due to increased demand for houses and simultaneously decreased savings by lifting borrowing constraints. Browning, Gortz and Leth-Petersen (2013) find evidence for this link between savings and house prices next to the aforementioned collateral effect. Moreover, Attanasio and Weber (1994) find that this mechanism is most likely to cause the correlation between house prices and consumption; one of their main findings is that younger households consume more than is expected based on theory after a house price boom, while this finding does not hold for older households. The authors argue that, given the financial markets’ circumstances in the sampling period, this observation is most in line with the financial liberalization hypothesis.
Two studies which examine the causal link between the synchronized movement of house prices and savings in the Netherlands are now discussed in more detail, since the present study also works with Dutch data. Even though both studies do analyze the same relationship between household savings and house price changes in the same country and period with alike methods, they still reach partially different conclusions.
Andreu (2014) studies the savings behaviour of Dutch households in a life cycle consumption model. According to his model, homeowners should correspond to an unexpected house price decline by increasing their savings, corresponding with the aforementioned wealth hypothesis. His model predicts
that this effect should become stronger the older the household is. However, by analyzing data of the Dutch Central Bank Household Survey, such an effect is not found. On the other hand, he does find that households increase their savings if they observe an actual decrease of aggregate house prices. He states that this finding can be explained best by the earlier mentioned collateral hypothesis.
Continuing, Beers, Bijlsma and Mocking (2015) find a negative effect of rising house prices on household savings. To find this, they use the same Dutch administrative panel data set which is used in the present study, containing slightly less than 2 million households. Among their findings is that households react to unexpected house price changes, which is supportive for the wealth hypothesis. However, this evidence is contradicted by the stronger reaction of young homeowners as compared to old homeowners, which is on its turn in accordance with the collateral hypothesis. Additional support for this hypothesis stems from their finding that households with a loan to value ratio greater than one reduce their savings more than those with a lower loan to value ratio.
All the above mentioned studies only derive a limited heterogeneity distribution of the effect of a house price shock, i.e. they only take into account the heterogeneity with respect to a few aspects of the reaction to changing house prices, while all relevant aspects should be studied simultaneously to come up with the correct effects. A peculiar and excellent exception to this is the work of Mian, Rao and Sufi (2013). They find large scale systematic evidence that there is substantial heterogeneity in the magnitude of the marginal propensity to consume out of increased housing wealth, which varies independently across income levels and degree of leverage. The present study also disentangles the underlying heterogeneity of the marginal propensity to save out of housing wealth in order to judge between the different causal explanations.
This summary of previous research may suggest that multiple processes are at play in the relationship between house price movements and savings. The different explanations are driven by different underlying theoretical models and different regression specifications, which are often not tested for misspecification of the functional form2. Surprisingly, none of the investigated
studies considers a more subtle dynamic specification of the model used to explain savings in presence of housing assets. Such a dynamic specification can be motivated empirically as well as theoretically, which is done in section 4. To the best of the author’s knowledge, this is the first study to come up with a dynamic model for the relationship under study, which makes the resulting findings a complement to the existing body of static models established by previous research.
2
3 The data
A substantive contribution of this study to the existing literature is the unique dataset to which all models are applied3. This section is devoted to why this
data set is so special. Moreover, it is motivated how this particular data set helps to gain an in-depth understanding of the relationship between house prices and savings and which modifications were necessary in order to do so.
The dataset used in this study is based on six administrative files from Statistics Netherlands. These six datasets are combined to get information on the household level about family composition, income, debts, (financial) assets and personal characteristics. The data is acquired from the Tax and Custom administration, the Civil registration, the Municipality administration and Dutch banks. The dataset is further expanded with relevant measures from Statistics Netherlands, such as measures for inflation and unemployment.
The variables used in the present study, including a detailed definition, are described in appendix A1. Availability of these variables on the household level can be seen as extraordinary on its own, since most previous research within this field lacks at least one of these variables important for a model explaining savings in presence of housing assets. In addition, the data used for this study is not derived from questionnaires, which make these data a lot more trustworthy than that of comparable studies4. Furthermore, sample selection
bias and related issued are not a problem in this study, because data of all
Dutch homeowners is available. Summarizing, this unique dataset enables the present study to leave out artificial corrections for selection problems5,
measurement problems and factitious imputations of hard to measure variables as house prices and savings. This ensures that the results are not an artefact of a combination between arbitrary technique choices and approximating important missing variables.
But even this dataset, despite its discussed advantages, is prone to errors of some form. The limitations and necessary fixes are discussed below.
Firstly, each time this study refers to the price of a house, it actually refers to the house’s WOZ-value, an administrative house price. The WOZ-value is determined by the market value of comparable sold houses last year. This means that the WOZ-value measures house prices with a lag of one year. Potential effects of this lagged measurement are prevented by regarding next year’s value as the house price of this year. In other words, the WOZ-value of a house in the year 2007 is determined by the price of similar sold
3
Readers should note that Beers, Bijlsma and Mocking (2015) were the first to use this unique dataset. Moreover, Mian, Rao and Sufi (2013) and Browning, Gortz and Leth-Petersen (2013) make use of an administrative panel data set, but not on the household level.
4
As mentioned before, the work of Browning, Gortz and Leth-Petersen (2013) and Mian, Rao and Sufi (2013) are exceptions to this. However, both studies do only contain house prices on the municipality level and impute consumption from a certain proxy, both contradictory to this study.
5
houses in 2006 and is therefore treated as the house price of 2006 in the used dataset. Moreover, according to Statistics Netherlands (2014) a house’s WOZ-value is highly correlated with the actual market WOZ-value of the house, which makes the WOZ-value a construct valid measure for house prices.
A second problem, which is encountered by merging the data sets, is the timing of measurement of the data. Some data are measured at the end of the year and are therefore representative for the situation of the household in the year after the measurement year, while other data is measured at the beginning of the year and is therefore representative for the measurement year. For each of the six data files, the measurement moment is taken into account to make sure the data is correctly matched.
Thirdly, the data on financial assets does not provide information about the type of mortgage the household possesses. Approximately 40% of the Dutch households had a savings mortgage in 2006 (Schilder & Conijn, 2012), because of the tax benefits described in appendix A4. A savings mortgage is not redeemed in parts during the loan period, but at once at the expiration date of the loan. To repay the loan at that date, savings are accumulated during the loan period. The dataset does not contain information about the magnitude of savings put aside to repay this type of loan. This is a severe limitation to the present study, as mortgage amount is overestimated and savings are underestimated using the available information, which inevitably causes measurement error effects on the estimated parameters.
Fourthly, this dataset does not distinguish between active savings by putting money aside on a savings account and passive savings by earning interest on the same bank account, while some related studies are capable to discern both forms of saving (e.g. Disney, Gathergood & Henley, 2010). However, it seems reasonable to assume that the results obtained are not driven by differences in interest rates on savings accounts, since the interest rate does not display a wide variation across banks during the sampling period.
Finally, errors in the data are fixed. For example, administrative mistakes in the mortgage data are discovered by inspecting the dataset for highly unlikely changes, defined as a change greater than plus or minus 20% of the mortgage amount in one year6. These administration errors are solved by
replacing the erroneous value with the mean mortgage amount of the past and subsequent year. Similar administrative problems as well as one year missing data are solved in the same way. Households with data that is not easily reconciled with usual behaviour, e.g. households that rent their house in one year and are the owner of the same house in the subsequent year, are removed from the data set7.
6
It can be argued that also some important observations are dropped, for example households that increased their mortgage for a home refurbishment. However, due to data limitations, these households cannot be recovered.
7
Also those households are perhaps wrongly deleted from the dataset, as there may be rational patterns for their behavior. However, again due to data limitations, recovery of these households is impossible. Besides, only a very small number of observations is deleted or modified in the above way (0.005% of the entire sample).
After execution of these households, the dataset must be made comparable to those of previous research within this area. In order to do so, all nominal monetary values are converted to real monetary values with 2006 as the base year using Dutch consumer price indices. Moreover, households that moved during the sample period are excluded. The reason for this is that moving is likely to affect both house prices and savings, but within a different causal framework than this study examines. In addition, households with unrepresentative house prices, defined as below 50000 euros or above one million euros are disregarded. The same is done with households in the 1% tails of the savings distribution. In the last elimination round, households which are not in the age range of 25 – 65 during all years of the sample period are removed. From the remaining records a balanced panel dataset is constructed. This results in a balanced panel with N = 1.458.686 households with all variables listed in appendix A1 available for T = 6 years8.
As will be explained in section 4 and 5, only the estimation results on municipality level can be trusted. In the models on municipality level, all the variables in table A1 are replaced with their municipality averages, i.e. real disposable income (𝑦𝑖,𝑡) is replaced with the mean real disposable income of all households living in a particular municipality m, i.e. 𝑦𝑚,𝑡= 1
∑ 𝐼𝑖∈𝑚∑𝑖∈𝑚𝑦𝑖,𝑡. It
can be argued that on the municipality level some of the above described eliminations are unnecessary. However, due to data limitations, recovery was not possible, except for the households outside the age range 25-65.
4 Methodology
The methodology section consists of four parts. Firstly, the estimation results of Beers, Bijlsma and Mocking (2015) (abbreviated BBM) are replicated and reanalyzed, as this study works with the same dataset as the present study. The analysis proofs the claimed necessity for a dynamic model. Secondly, it is argued that the dynamics are appropriately modelled by the auto-regressive distributed lag (hereafter ARDL) model. It is demonstrated that this model implicitly takes into account both short run and long run dynamics, without precluding the model currently used in BBM. Thirdly, the employed estimation method is discussed, but for details the reader is referred to appendix A3. The section ends with describing the models on which inference conducted in the results section is based.
4.1 Reanalyzing the model of Beers, Bijlsma and Mocking
(2015)
As a starting point, the preferred model specification of BBM is re-estimated. Their preferred regression specification is given in equation (1). Replication of
8
I thank Remco Mocking, Nancy Beers and Michiel Bijlsma for carrying out a great part of the fixes and eliminations discussed in this section.
their results is done using a random sample of 25% of the original data set due to computational challenges associated with parameter estimation using such a large data set9. Estimation is carried out with their preferred fixed effects panel
data estimator.
𝛥𝑠𝑖,𝑡 = 𝛼𝑖+ 𝜃𝑡+ 𝛽∆ℎ𝑖,𝑡+ 𝜸∆ℎ𝑖,𝑡𝒛𝒊,𝒕+ 𝜹𝒛𝒊,𝒕+ 𝝋𝒙𝒊,𝒕+ 𝜀𝑖,𝑡 (1) i = 1…N t = 1…T
In model (1), 𝛥𝑠𝑖𝑡 presents the change in savings of household i for period t, 𝛼𝑖 is the household specific effect, 𝜃𝑡 represents the time dummies, ∆ℎ𝑖,𝑡 denotes the one year house price change, 𝒙𝒊,𝒕 is a vector containing the control variables income, regional unemployment, leverage, household size, source of income dummies, marital status dummies and a dummy for presence of children, as defined in table A1. 𝒛𝒊,𝒕 contains variables which are interacted with first differences of house prices: age, the underwater dummy, the declined house price dummy and a dummy whether the most important source of income for a household is entrepreneurship.
As expected, the (unreported) estimation results of model (1) almost exactly match with the estimation results given in BBM. Most coefficients are the same up to three digits and the 95% confidence intervals of each coefficient contains the point estimate of that regressor as reported in BBM. In addition, the standard errors of the replicated estimates are approximately two times as large due to the 25% random sample. This indicates that the forthcoming results are neither driven by the random sample, nor by a different interpretation of the variables in the data set.
Unlike the authors of the aforementioned study, the present study does check model (1) on particular forms of misspecifications of the functional form. The results are rather striking.
According to the Wooldridge test (Wooldridge, 2010), residual autocorrelation is present in the residuals (F(1, 362269) = 1539.90, p = 0.001 for the null hypothesis of no serial autocorrelation in the residuals). However, the Wooldridge test is determined after estimating equation (1) using a first differences transformation instead of the appropriate within transformation. Therefore, it is questionable whether this test should be trusted. Another test which indicates the presence of a first order autoregressive structure in the residuals is the Bhargava, Franzini and Narendranathan Durbin Watson statistic (Bhargava, Franzini & Narendranathan, 1982). A rule of thumb states that if this statistic exceeds 2, this clearly indicates that autocorrelation is present in the residuals of the original fixed effects estimates for N larger than 1000. The value of this statistic is 2.12 and hence absence of autocorrelation is
9
A fixed effects regression on the full sample takes about 1.5 hours and due to limited available memory estimation with the full sample is not always possible.
rejected. Summarizing, strong signs exist that the residuals of specification (1) display an autoregressive structure. This indicates that specification (1) suffers from neglected dynamics, omitted variables or a wrong functional form, resulting in biased coefficient estimates and uninterpretable standard errors.
Despite the fact that the last paragraph already has falsified the static specification of model (1), a test for the correct functional form conditional on this static specification is carried out. This is done with the Ramsey RESET test (Ramsey & Schmidt, 1976) and is essentially an auxiliary fixed effects regression of specification (1), with squared model fit added to the right hand side of the regression equation. If the model is well specified, there is no need to include the squared model fit as a regressor and hence it should not have significant explanatory power. However, a highly significant coefficient turns up
(t(1811497) = 2.42, p = 0.02, two-tailed), potentially indicating that cross
terms and squared regressors are wrongly omitted in the preferred specification of BBM10.
Continuing, model (1)’s specification of house prices in first differences including the interactions with the first differenced house price variable are put to a formal test. It is easily derived that incorporating the house price variable in first differences boils down to estimating regression specification (1’), while imposing the restriction 𝛽0= −𝛽1 and 𝜸𝟎= −𝜸𝟏 (for each element of this vector).
𝛥𝑠𝑖,𝑡 = 𝛼𝑖+ 𝜃𝑡+ 𝛽0ℎ𝑖,𝑡+ 𝛽1ℎ𝑖,𝑡−1+ 𝜸𝟎ℎ𝑖,𝑡𝒛𝒊,𝒕+ 𝜸𝟏ℎ𝑖,𝑡−1𝒛𝒊,𝒕−𝟏+ 𝜹𝒛𝒊,𝒕+ 𝝋𝒙𝒊,𝒕+ 𝜀𝑖,𝑡
i= 1…N t = 1…T (1’)
These restrictions are tested by an F-test and are strongly rejected for house prices (F(1, 362341) = 42.44, p = 0.001) as well as for the interaction between house prices and age (F(1, 362341) = 56.97, p = 0.001) and for the interaction with the declining house price dummy (F(1, 362341) = 5.68, p = 0.05 ). For the interaction with the negative equity dummy the restriction is not rejected (F(1,
362341) = 1.78, N.S.), which is also the case for the interaction with the
entrepreneur dummy (F(1, 362341) = 0.8794, N.S.). However, testing all restrictions simultaneously clearly rejects the specification in first differences
(F(5, 362341) = 22.78, p = 0.001) and thus shows that specification (1) is
unnecessarily restrictive.
Another restriction which should be put to a formal test is the specification of the dependent variable in first differences. This restriction could in principle be tested in the same way as is done in the last paragraph.
10
Although this test is originally meant for ordinary least squares, in the present study it is also safely applied to a regression equation estimated with the fixed effects estimator, because it can be shown that fixed effects is essentially the same as least squares dummy variable regression, which is appropriately estimated with OLS.
However, this means that a lag of the dependent variable enters the right hand side of model (1), i.e. model (1’’) must be estimated:
𝑠𝑖,𝑡 = 𝜆𝑠𝑖,𝑡−1+ 𝛼𝑖+ 𝜃𝑡+ 𝛽∆ℎ𝑖,𝑡+ 𝜸∆ℎ𝑖,𝑡𝒛𝒊,𝒕+ 𝜹𝒛𝒊,𝒕+ 𝝋𝒙𝒊,𝒕+ 𝜀𝑖𝑡 (1’’)
i= 1…N t = 1…T
If the specification in (1’’) is estimated with fixed effects, the resulting estimate of λ will not be consistent, since the lagged dependent variable is predetermined with respect to the error term in equation (1’’). Additional danger arises because testing the restriction that λ = 1 should be done with a unit root test. As a consequence, this test is not performed, but the thoughts about this test together with the aforementioned test results have brought about a final model, to be discussed in section 4.4, which does not impose any restrictions on the coefficients and thereby neither precludes nor imposes the former specification of model (1).
Now that it is shown that the current model’s specification in first differences is too restrictive, another question naturally arises. Since the house price variable does not take on negative values, contradictory to its first differenced counterpart, it could also be incorporated in natural logarithms instead of levels. This log transformation is in line with most similar research within this field. Therefore, equation (1) is re-estimated two times: one time the house price variable and its interactions are specified in levels and one time in logs. Which of the two specifications is most appropriate11, is investigated
with the Davidson-MacKinnon J-test for non-nested models (Davidson & MacKinnon, 1981). This test examines whether the model fit of regression equation (1) with house prices specified in logs adds explanatory power to regression equation (1) with house prices specified in levels and vice versa. As can be seen in table 1, the Davidson-MacKinnon J-test is inconclusive about which specification is more appropriate12. Therefore, the result of this test is
not interpreted as misspecification of the functional form of equation (1), but merely as an indication that different specifications of the house price variable are possibly correct.
11
Not to be confused with correct, regarding the results of the misspecification tests of this model.
12
Note that the original Davindson-MacKinnon J-test is meant for models estimated with OLS. However, as mentioned before, the fixed effects estimator is essentially the same as the least squares dummy variables estimator, which is a special form of the OLS estimator. Therefore, it seems logical to assume that this test keeps the properties established in MacKinnon (1983).
Table 1
Results Davidson-MacKinnon J-test
Notes. Estimation is carried out using the fixed effects estimator. P-value is with respect to two-sided alternative.
The same test could be carried out for the disposable income variable. However, this variable does take on negative values for a very small fraction of the sample (0.0022%). Moreover, if savings is not incorporated in first differences but in levels, this variable can also be log transformed. But also the savings level is negative for a very small fraction of the sample (0.0033%). Fortunately, this problem is solved if the data is aggregated on the municipality level, as is done for the final model. Besides, it appears both empirically and theoretically sound to take logarithms of the savings, disposable income and house price variable and therefore this transformation is employed.
Thus, taken together, reanalyzing the estimation results of BBM has resulted in the following insights for the development of a new model. Firstly, tests for residual autocorrelation indicate that the residuals display an autoregressive structure. This is a clear sign that the current restricted static model must be replaced by an unrestricted dynamic model. Secondly, the significant coefficient of the squared model fit when added to specification (1) demonstrates that more interactions than included might play a role in the relationship under study and therefore a more adequate number of interactions, determined by a procedure outlined in section 4.4, will be added to the model. Thirdly, coefficient restriction tests exhibit that restrictions imposed in model (1) are violated. Consequently, a more flexible model which does not preclude the restrictions of model (1) but does neither impose them will be used. Fourthly, the Davidson-MacKinnon J-test reveals that a specification of non-negative monetary values in logs should be considered. Accordingly, all models will be estimated using logarithms of savings, disposable income and house price variables, as this appears to be both theoretically and empirically sound.
4.2 The auto-regressive distributed lag model
The analysis in the previous section reflects that the static model of regression specification (1) suffers from neglected dynamics. The implicit underlying assumption of such a static model is that all effects from changes of the regressors are completed within the period the change occurs. In other words, the effect of a changed house price on savings must immediately be fully realized. Subsequently, if none of the regressors in (1) changes within a time period, the dependent variable must remain unchanged as well. Hence, it is
Original model House prices in logs House prices in levels
Added model fit Levels Logs
P-value added fit p = 0.001 p = 0.001
assumed that the system is always observed in its equilibrium. This assumption is highly restrictive and often violated by real world empirics.
In a linear dynamic model lags of the contemporaneous regressors as well as lags of the dependent variable are added to the right hand side of the regression equation. This is theoretically sound according to two hypotheses (Harvey, 1981). Adding lags of the contemporaneous regressors is in line with the adaptive expectations hypothesis, which states that decision makers revise their expectations based on the difference between their expectations and the realized values of the contemporaneous regressors. Adding lags of the dependent variable is in line with the partial adjustment hypothesis, which states that because of rigidities in behaviour, only a partial adjustment is possible in the current period: the remainder of the change will be completed in the following periods13.
Both hypotheses seem plausible in a model explaining savings in presence of housing assets. Homeowners may have expectations about the price of their house in the future and revise those expectations based on realized house prices. Since this realization is not directly observable, it seems reasonable to assume that assimilating this realization happens gradually. Furthermore, households may wish to increase their savings because of decreased housing wealth. However, rigidities in their consumption and savings pattern may prevent households to realize such additional savings immediately. In other words, it may take time to adjust to a new set of saving habits induced by the decreased housing wealth. It is therefore realistic to assume that households only move partly away from their current savings position, while the remainder of the change is completed in future periods.
The general regression specification motivated by the aforementioned hypotheses as well as by not imposing any restrictions on the functional form is given in (2).
𝑠𝑖,𝑡 = 𝛾𝑠𝑖,𝑡−1+ 𝛽0𝑥𝑖,𝑡+ 𝛽1𝑥𝑖,𝑡−1+ 𝜖𝑖,𝑡 (2)
i= 1…N t = 1…T
In (2) 𝑠𝑖,𝑡 denotes the household’s level of savings, 𝑠𝑖,𝑡−1 denotes the level of savings of the previous period, 𝑥𝑖,𝑡 is a contemporaneous regressor and 𝑥𝑖,𝑡−1 is the one period lagged regressor14.
The specification in (2) can be classified as an autoregressive distributed lag model. This model is chosen because it incorporates both short run and long run dynamics in the way described in the previous paragraph. In addition, both the long- and short term effect of a change of one of the regressors can be
13
Note that the coefficient of the lagged dependent variable can also appropriately model non-rigidities or extreme switching behavior.
14
Note that this specification can be generalized to vectors of regressors and more lags of both the dependent variable and the regressor, but a scalar example using one lag is selected for the ease of exposition.
derived without estimating the long-run relationship explicitly. To see this, equation (2) must firstly be rewritten as follows:
𝛥𝑠𝑖,𝑡 = (𝛾 − 1)𝑠𝑖,𝑡−1+ 𝛽0𝛥𝑥𝑖,𝑡+ (𝛽1+ 𝛽0)𝑥𝑖,𝑡−1+ 𝜖𝑖,𝑡 (2’)
i= 1…N t = 1…T
In equation (2’) the long-run effect of a change in 𝑥𝑖,𝑡 can easily be derived from the equilibrium condition. If the system is in equilibrium, this means that 𝛥𝑠𝑖,𝑡 = 0 and 𝛥𝑥𝑖,𝑡= 0. Moreover, in equilibrium 𝑠𝑖,𝑡−1= 𝑠𝑖,𝑡 = 𝑠𝑖∗ and
𝑥𝑖,𝑡−1= 𝑥𝑖,𝑡= 𝑥𝑖∗ must hold. Therefore, in equilibrium, equation (2’) can be
rewritten as 𝑠𝑖∗=𝛽1−𝛾1+𝛽0𝑥𝑖∗
, which determines the long run effect of a unit change in 𝑥𝑖,𝑡 as 𝛽1+𝛽0
1−𝛾 . If this long run effect is renamed λ and is substituted
back in (2’) the following expression is derived after some manipulation:
𝛥𝑠𝑖,𝑡 = 𝛽0𝛥𝑥𝑖,𝑡− (1 − 𝛾)[𝑠𝑖,𝑡−1− λ 𝑥𝑖,𝑡−1] + 𝜖𝑖,𝑡 (2’’)
i= 1…N t = 1…T
Equation (2’’) has an interesting interpretation. Firstly, if γ=1, i.e. if the savings process contains a unit root15, the distance from the equilibrium does
not play a role in determining the effect of a shock in 𝑥𝑖,𝑡 and hence there does not exist an equilibrium in this relationship. Secondly, if 𝑠𝑖,𝑡−1 is above its’ equilibrium value, in other words 𝑠𝑖,𝑡−1 > 𝑠𝑖∗ , the term in brackets will be positive: to adjust the positive deviation from the equilibrium level, an error correction has to take place which occurs if γ<1. Summarizing, the effect of a unit change in 𝑥𝑖,𝑡 is determined by the short term effect 𝛽0 and the deviation from the long run equilibrium in the previous period −(1 − 𝛾)[𝑠𝑖,𝑡−1− λ 𝑥𝑖,𝑡−1]. Separate effects for each period are derived in appendix A2.
4.3 Estimation method
The original ARDL model is suitable for the analysis of time series. However, the data of this study is in panel data format. Fortunately, this model is easily extended to fit to a panel data context and the Arellano Bond dynamic panel data estimator can be applied to estimate the coefficients of the ARDL model.
15
Because an estimator suitable for small T large N is used in the current study, the relied on asymptotics is with respect to N and not with respect to T. This means that presence of a unit root, which is highly unlikely in the used framework and rejected by the results presented in section 5, does not immediately change the analysis.
Using panel data rather than pure time series data to estimate the relationship between savings and house prices has several advantages. Firstly, the cross household variation in savings and house prices is exploited next to the within household over time variation of these variables. This results in substantially augmented variability of the data compared to merely considering either the cross section dimension or the time series dimension. Secondly, the panel data estimation method removes the bias of the estimated parameters caused by a household specific effect (municipality specific effect for the data aggregated on municipality level). Thirdly, the used dynamic panel data estimator can deal with endogenous regressors, predetermined regressors and exogenous regressors, which is extensively discussed in appendix A3.
Four different versions of the Arellano Bond estimator are used within this study. The estimators differ in the used weighting matrix and the used instrument matrix. The results generated using the one (two) step weighting matrix are referred to as AB1 (AB2). Sensitivity with respect to the used weighting matrix is taken into account because of an involved bias efficiency trade-off: the AB2 estimator is asymptotically efficient, but simulation has shown that this is mitigated with larger finite sample bias (Kiviet, Pleus and Poldermans, 2014). The abbreviation is expanded with the letter F (C) if the full (collapsed) instrument matrix is used. Sensitivity with respect to the used instrument matrix is taken into account since using a collapsed instrument matrix results in a Hansen test, its importance being discussed below, which has reasonable size control, while this seems less true for estimates using the full instrument matrix. In addition, collapsing reduces finite sample bias, but with the price of slightly deteriorated precision (Kiviet, Pleus and Poldermans, 2014). Furthermore, Roodman’s (2009) advice is followed to check whether estimation results are robust to reducing the instrument count, which is what collapsing essentially does. Details about these four estimation methods are discussed in appendix A3.
Each model’s estimation results are accompanied by the AR(1)-test, AR(2)-test and Hansen AR(2)-test to judge to which extent the estimated parameters are reliable. These tests are further discussed in appendix A3. As Kiviet, Pleus and Poldermans (2014) note, both size and power of these tests may be highly distorted and hence judging their results should be carried out with great caution. For this reason, trustworthiness of the estimation results is also evaluated based on alternative criteria. Firstly, the variance of the entity specific effects is compared with the variance of the residuals, because in a powerful model the former is not much larger than the latter. Secondly, the estimated long term income elasticity with respect to savings is computed, which according to the literature should be between zero and one (see for example Mankiw and Taylor, 2008). A model that is unable to estimate this elasticity inside its credible range can neither be trusted to produce reliable estimates for the housing wealth effect. Thirdly, Bond (2002) has identified a bracketing range for the coefficient of the lagged dependent variable: its’ coefficient should be between the downwardly biased fixed effect estimate and
the upwardly biased ordinary least squares estimate. Fourthly, the induced dynamic pattern should be similar for the different employed estimators, as they are used to estimate the same underlying process. These four alternative evaluation criteria for reliability of each model’s estimates will accompany the aforementioned panel data misspecification tests.
A final note must be made about the standard errors used in this study. The usual standard errors are way too optimistic for the several versions of the Arellano Bond estimator employed in this study. Hence, inference is based on heteroskedasticity consistent standard errors for the AB1 estimates and Windmeijer corrected standard errors for the AB2 estimates (see Windmeijer, 2005). Standard errors for any (non)-linear combination of the estimated coefficients, for example the effects derived in appendix A2, are computed with the delta method.
4.4 Regression equations
Now that a suitable modelling framework (ARDL model) and estimator (Arellano Bond estimator) are derived, it is time to discuss the estimated models. Pretesting has shown that the form of regression specification (3) has the greatest empirical support for estimation of the relationship between house prices and savings given the number of time periods available; expanding regression equation (3) with additional lags of the dependent and independent variable results in insignificant coefficients for all variables while tests for residual autocorrelation have shown that adding these lags does not affect the autoregressive structure in the residuals. This results in the following basic model:
𝑠𝑖,𝑡 = 𝛾𝑠𝑖,𝑡−1+ 𝛼0ℎ𝑖,𝑡+ 𝛼1ℎ𝑖,𝑡−1+ 𝛿0𝑦𝑖,𝑡+ 𝛿1𝑦𝑖,𝑡−1+ 𝜷𝟎′𝒙𝒊,𝒕+ 𝜇𝑖+ 𝜃𝑡+𝜖𝑖,𝑡 (3)
i = 1…N t = 1…T
In model (3) savings level for household i in time period t (𝑠𝑖,𝑡) depends on one period lagged savings (𝑠𝑖,𝑡−1), house prices in the current period (ℎ𝑖,𝑡), house prices in the previous period (ℎ𝑖,𝑡−1), disposable income in the present period (𝑦𝑖,𝑡), disposable income in the previous period (𝑦𝑖,𝑡−1), a vector of contemporaneous control variables (𝒙𝒊,𝒕) and an individual effect 𝜇𝑖 next to a time period specific effect 𝜃𝑡. The monetary amounts (𝑠𝑖,𝑡, ℎ𝑖,𝑡 and 𝑦𝑖,𝑡 and their lags) are all log transformed, because this enhances coefficient interpretability and is in addition sound from a theoretical point of view (see for example Disney et al., 2010 and section 4.1 of the present study). The set of control variables (𝒙𝒊,𝒕) is derived from a combination of theory, previous research and data availability. Pretesting, from which results are available upon request, has resulted in including the following variables in 𝒙𝒊,𝒕: disposable income, regional unemployment, leverage, household size, source of income dummies, marital
status dummies, a dummy for declined house prices, the underwater dummy and a dummy for presence of children, all exactly defined in table A1.
Unreported results show that the data strongly rejects the orthogonality conditions necessary to ascertain consistent estimates, even if all regressors, except the time dummies, are classified as endogenous with respect to the error term in equation (3). Moreover, second order autocorrelation is present in the residuals, while it should be absent. Furthermore, these two unpleasant findings are invariant to the used estimation method. Two explanations for this are considered. On the one hand, the findings of Kiviet, Pleus and Poldermans (2014) suggest that both the Hansen test and AR(2) test are size distorted and in some situations do reject the instruments and absence of second order autocorrelation, while the reverse is true. Support for this argument is further augmented by the finding that the difference in Hansen test rejects the validity of the time dummies as exogenous instruments, while this is true by construction. The same happens for the lagged savings variable, which is predetermined with respect to the error term in equation (3), but rejected as such by the incremental J-test. On the other hand, a more likely explanation is that the conditions for consistency of the coefficient estimates are truly not satisfied, because of inadequacies in model (3)’s specification: this model aims to explain household savings by 24 regressors in total, while many more processes may be at play at this highly non-aggregated level. For example, the model does not take into account the purchase decision of (expensive) durable goods, e.g. a car, while this certainly affects the savings level of the household. This may be one of the causes for the autoregressive structure in the residuals and hence also for rejection of the orthogonality conditions. Another cause is that model (3) imposes the same dynamics within the relationship under study to all households, while it seems appropriate to assume that the dynamics is asymmetric among subgroups of the population. The problem might be exacerbated by the large power of the Hansen test when the cross section dimension of the data is large: if a small part of the households behaves completely different than the model supposes, this results in residual autocorrelation so large in magnitude that the moment conditions are rejected for all households, whilst for a large part of the households the conditions are actually satisfied.
An unlikely but potential cure for the invalidity of the moment conditions might be that interactions need to be added to model (3). As is described in section 4.1, interactions between regressors seemed to be wrongly left out in the original model used by BBM. Therefore, an extension of model (3) includes cross terms between house prices and other regressors. This results in model (4) in which the interactions are contained in the vector 𝒛𝒊,𝒕.
𝑠𝑖,𝑡 = 𝛾𝑠𝑖,𝑡−1+ 𝛼0ℎ𝑖,𝑡+ 𝛼1ℎ𝑖,𝑡−1+ 𝛿0𝑦𝑖,𝑡+ 𝛿1𝑦𝑖,𝑡−1+ 𝜇𝑖+ 𝜃𝑡+ 𝜷𝟎′𝒙𝒊,𝒕+
𝜷1′𝒙𝒊,𝒕−𝟏+ 𝝆𝟎′𝒛𝒊,𝒕+ 𝝆𝟏′𝒛𝒊,𝒕−𝟏+ 𝜖𝑖,𝑡 (4)
Including interactions boils down to making a lot of choices, because all regressors could potentially be interacted with the house price variable and still their effect may have a meaningful interpretation. Hence, to decide which interactions should be included in the vector 𝒛𝒊,𝒕 a top down approach has been employed. Firstly, the house price variable is interacted with all contemporaneous regressors of equation (3). In the next step, each interaction appearing in model (4) with an absolute t-ratio smaller than one for both the contemporaneous coefficient as well as the lagged coefficient is eliminated. This procedure is repeated until no interactions from which both the lag and the contemporaneous coefficient have an absolute t-ratio smaller than one appear. For reasons of model parsimony, interactions with the source of income dummy variables and with the marital status dummy variables are disregarded, although the final set of included interactions appears relatively insensitive to taking them into account. Repeating this procedure for all four estimation methods has resulted in interacting disposable income, loan to value ratio, age, leverage, household size and a declined house price dummy with house prices, all contained in the vector 𝒛𝒊,𝒕 and their lags in the vector 𝒛𝒊,𝒕−𝟏
Admittedly, the above statistical procedure is rather mechanical and whether this results in the right set of interaction variables cannot be formally tested. Nevertheless, according to the present study, this procedure is at most as arbitrary as only including interactions claimed by economic theory, because then statistically important interactions are easily overlooked16. Moreover,
previous research did not explicitly show any misspecification results of their models and hence it cannot be assessed whether their included interactions are potentially driven by omitting important other interactions17.
However, as unreported results show, including these interactions does not solve the issues raised by the violated moment conditions and by the presence of second order autocorrelation in the residuals of equation (3). The same argumentation as in the last section is followed; a model with 35 regressors is just too restrictive for explaining household level savings. This argument is further reinforced by the large standard deviation of the individual effects, ranging from 0.5831 to 1.905, especially compared to the relatively small variance of the residuals, ranging from 0.2744 to 0.2929. This elicits that a large part of the variance in savings is not adequately explained by the regressors in model (4) and therefore ends up in the individual specific effects, which is interpreted as a clear sign that model (4) falls short for the present
16
However, the reverse may also happen: economically interesting variables which don’t have a significant impact are excluded, while this is of course informative.
17
It is acknowledged that also interactions with regressors different than house prices are important, but pretesting has shown that this neither enhanced model fit nor resulted in a set of interactions with a t-value greater than 1 proposed by the different estimation methods: the repetitive procedure had eliminated all interactions in the end. An exception to this are the interactions with income; adding these is used as a robustness check in section 5.6.3.
limited set of available data regarding special circumstances for the individuals involved.
Summarizing, the above models (3) and (4) do not pass the tests for validity of the moment conditions necessary to ensure consistent parameter estimates, regardless the used estimator. It is motivated that this may be due to 1) omitted variables in equation (3) and (4) resulting in a complex autoregressive structure in the residuals 2) distortion of the size of the misspecification tests and 3) the enormous heterogeneity in savings behaviour across households. In order to partly overcome these problems, the above analysis is repeated on municipality level. Aggregating the data over municipalities hopefully averages out the influence of household specific decisions which cannot be taken into account in the above restrictive models (additional arguments for aggregating data are given by Blundell and Stoker, 2005). This results in model (5), which is the municipality equivalent of (3), and model (6), which is the municipality equivalent of (4).
𝑠𝑚,𝑡= 𝛾𝑠𝑚,𝑡−1+ 𝛼0ℎ𝑚,𝑡+ 𝛼1ℎ𝑚,𝑡−1+ 𝛿0𝑦𝑚,𝑡+ 𝛿1𝑦𝑚,𝑡−1+ 𝜷𝟎′𝒙𝒎,𝒕+ 𝜇𝑚+ 𝜃𝑡+𝜖𝑚,𝑡 (5) (5) m = 1…M t = 1…T 𝑠𝑚,𝑡= 𝛾𝑠𝑚,𝑡−1+ 𝛼0ℎ𝑚,𝑡+ 𝛼1ℎ𝑚,𝑡−1+ 𝛿0𝑦𝑚,𝑡+ 𝛿1𝑦𝑚,𝑡−1+ 𝜷𝟎′𝒙𝒎,𝒕+ 𝜇𝑚+ 𝜃𝑡+ 𝝆𝟎′𝒛𝒎,𝒕+ 𝝆𝟏′𝒛𝒎,𝒕−𝟏+ 𝜖𝑚,𝑡 (6) m = 1…M t = 1…T
In model (5) and (6) the variables of equation (3) and (4) are replaced with their appropriate municipality means, as explained in section 3. Because the data is now aggregated on the municipality level, it is no longer possible to conduct inference on the household level. Moreover, the crossection dimension of the data is extraordinarily decreased: from 1.458.686 households to 402 municipalities. Nevertheless, these disadvantages are easily reconciled with the advantage of valid inference in a particularly interesting range of values of each variable and, as will be discussed later, with the advantage of passing the misspecification tests in order to argue that the estimated coefficients may be consistent.
To estimate the coefficients of equation (5) and (6) with the Arellano Bond estimator, it is necessary to divide the regressors in three subsets of endogenous, predetermined and exogenous regressors with respect to the error terms of equations (5) and (6), as is discussed in appendix A3. This is done by a combination of the difference in Hansen test and conventional wisdom18.
18
Guidance solely on the difference in Hansen test is not possible, because classifying all regressors as endogenous and thereafter adding moment conditions trying to reach an instrument set which is not rejected gives ambiguous results.
Regressors from which it seems highly unlikely that they are involved in a reverse causality problem within the structural equation are classified as exogenous. These are: the marital status percentages, the source of income percentages, presence of children percentages, regional unemployment, household size, the declining house price percentages and the time dummies. As can be seen in table 2, the implied moment conditions by classifying these regressors as exogenous are not rejected by the incremental J-test, which, apart from type 2 errors and size distortion of this test, confirms the exogeneity assumption. Regressors from which it is difficult to tease apart the direction of causality with savings are leverage, loan to value ratio and the underwater dummy19. As emerges from table 2, the moment conditions implied by
classifying these variables as predetermined are not rejected and hence these regressors are treated as such20. Furthermore, it may be doubtful how
disposable income and house prices, including its interactions, must be classified. Classifying these regressors as exogenous is only correct if feedback effects of these variables on savings can completely be ruled out. However, lagged feedback or even instantaneous feedback may play a role, by which these regressors should be treated as predetermined or endogenous respectively21. The difference in Hansen test reported in table 2 shows that the
moment conditions implied by treating these regressors as predetermined are not rejected and hence these regressors are classified as such. The possibility that the aforementioned variables are falsely classified as predetermined is examined as a robustness check in section 5.4.3. The classification is summarized in table 2.
The difference in Hansen test seems to work after “starting it up” with some strong and valid instruments.
19
For example, a high loan to value ratio may urge the household to save more in order to have a buffer for future financial setbacks. Reversely, a high level of savings may be a strong buffer against a high loan to value ratio. A similar reasoning can be constructed for the other regressors in this subset.
20
Which does not mean that these variables are in fact predetermined with respect to the error term, because of the reasons mentioned before.
21
For example, a high disposable income may result in high savings but reversely a high level of savings might make it possible to structurally decrease working hours and thereby disposable income in upcoming years. In addition, disposable income might even be endogenous with respect to the error terms in equation (5) and (6) because disposable income and savings are both partly jointly determined by the manner by which households save for retirement, which is omitted in the current model. Fortunately, the identification strategy stories can be judged based on empirics by the incremental J-test.
