Modelling household spending using a random assignment scheme

Consumer spending in the UK amounted to 872 billion pounds in 2009 (ONS, 2010). It is understandable therefore, that both commercial and public organisations have an interest in gaining a better understanding of this sector of the economy. In the private sector, this might be to predict the size of the market for particular goods or services. For governments, it is important to understand the effect of indirect taxes that affect households differently depending on the type and quantity of goods they consume. The problem this paper addresses is how to model the way spending on various goods and services varies in response to demographic, economic and socio-technical change.

In microsimulation, as well as in economics generally, modelling household expenditure is usually carried out by econometric methods. However, there are some difficulties with this approach such as the amount of data needed to estimate the parameters accurately (Thomas, 1987) and the difficulty of modelling at a highly disaggregated level (discussed below). This paper examines the use of what are known as random assignment schemes as a way to model household expenditure. This approach is based on the idea of predicting the behavioural response of a microsimulation unit by finding a donor, which is in some sense similar to the receiving unit. According to Klevmarken (1997), the advantages of this method are that it is not necessary to impose a functional form on the data or make any assumptions about the distribution of variables. There are no parameters to estimate and the method preserves the variation and most of the correlation present in the original dataset. The approach also allows the study of situations where people behave in fundamentally different ways; in particular where some individuals do something other than maximise their utility function.

Following a brief review of econometric modelling, the paper introduces the principles of random assignment schemes. These are explicated further in an example model to predict the effect of changes in the level of income on household expenditure patterns. The results are validated by showing that the model reproduces some stylised facts that are already known about household expenditure patterns. The model is then used as a platform to show how the random assignment scheme can be used to model a large number of goods, at the level of individual households.

In economics, the standard approach to modelling household demand is by using econometric methods. It is possible to do this in a single, regression type equation of the form.

Here, the dependent variable Y might represent the budget share for food. The independent variables X₁ to Xn could represent factors that are thought to influence spending on food such as household size, income and price. The constants b₀ to bn would be estimated using standard statistical software on observed data that captures the relationship between the relevant variables.

One of the problems with this approach is that it is necessary to specify a separate equation for each good of interest. This becomes unwieldy if the number of goods is large as it would be in a typical household budget set. It is also difficult to model the interaction between spending on each good because, in principle, this will depend on what is spent on all the other goods. The list of independent variables should then include the budget shares of all these items. This is feasible for a small number of goods but as the size of the budget set increases, the number of parameters needed to estimate the model grows quickly to the point where, for most datasets, there are not enough cases to provide accurate estimates of the parameters.

This problem is alleviated to some extent by the use of complete demand systems consisting of an integrated set of equations. One of the most sophisticated is the Almost Ideal Demand System (AIDS, Deaton and Muellbauer, 1980). It uses the principles of neoclassical economic theory to impose restrictions on the possible values of the parameters and so reduce the amount of data needed to estimate them. The AIDS model is used here as a representative of the econometric approach, partly because it may be the most advanced (Alpay & Koc, 1998) and because it seems to be one of the most widely used.

The general model for the Almost Ideal Demand System, for a budget set of i goods is:

where

• w_i is the budget share of the i^th good

• M is the total consumption expenditure

• P_j is the price of the j^th good (j is a good other than i)

• P is a price aggregator for the set of goods

• ε_i is an error term for good i

It is possible to take the demographic characteristics of each household into account by including a vector of dummy variables Z.

The dummy variables indicate the presence or absence of the characteristic of interest. Income, for example, could be divided into a number of bands and each household would have a 1 if it is in a particular band and a 0 otherwise. In this way, there is a separate equation, with its own parameters, for each income band.

It can be seen from the γ_ij and the δ_iz that the number of parameters to estimate increases with the square of the number of goods and as the product of the number of household categories and goods. As a result, this approach is limited to consideration of a relatively small number of goods and household types. It becomes more difficult to apply if the households are to be represented at a highly disaggregated level as they are in microsimulation modelling. Here, as the number of dummy variables increases, the number of parameters becomes prohibitive due to data limitations. Also, the number of income bands is limited by the number of equations in the system so it would not be possible to use continuous variables.

The difficulties associated with parametric estimation of demand systems raises the question of whether there are alternative methods that do not involve parameters. Random assignment provides the basis for one such method. The idea of random assignment is usually associated with selecting individuals for treatment groups in such a way that the effect of the treatment is the only source of difference in outcomes between the groups. However, in the context of microsimulation modelling, random assignment is a kind of matching or imputation technique where a donor is selected on the basis of its similarity or closeness to the receiving unit. Klevmarken (1997) provides an example of how a random assignment scheme can be used as a method of projecting a variable over time. Data is available on a set of incomes for two consecutive years along with some related variables such as age and sex. It is desired to project the income distribution for the following year. This is done by first defining a distance metric between a donor’s variables in year 1 and a receiver’s variables in year 2. The essence of the procedure is that the income for each case in year 3 is obtained by finding a donor, whose characteristics in year 1, are similar to those of the current case in year 2. The receiver’s income in year 3 is then assigned to be what the donor’s income subsequently became in year 2.

Random assignment has been used by (Klevmarken et al., 1992) and (Klevmarken & Olovsson, 1996) and more recently by (Holm, Mäkilä, & Lundevaller, 2009) in a dynamic spatial microsimulation model of geographic mobility. They found that this approach had the potential to provide better population projections than the alternative interaction based models. However, they also noted that the representation of behaviour is limited to what has already been observed in the initial data set. This is not a problem when it is desired only to project, all other things being equal, from current data but it is a limitation when applying the method in new situations because the behaviour and correlation structure are locked in to what has been observed. However, the issue of how to extrapolate from observed data is common to all approaches. In microsimulation, this is often done by applying some kind of alignment procedure. It would also be possible to use theoretical assumptions or empirical data to extend the model.

The next part of this paper introduces a simple example application of random assignment to model the effect of changes in household income on household expenditure patterns. This shows the operation of random assignment in more detail.

Economic modelling is often carried out at the individual level. This makes sense because it is the individual who makes decisions and has some agency regarding their consumption behaviour. However, it is possible for individuals to have no income of their own yet spend money on a range of items. This is explainable by intra-household allocation of resources and at the individual level, this would have to be represented in the model. Working at the household level encapsulates intra-household allocations implicitly in observed spending patterns and so simplifies the specification of the model.

4.4 Results

4.4.1 The average household

When the initial data file, taken from the 2006 EFS, was read in, mean household income (before tax and including any benefits) was £656. This was increased in a series of 10 steps (10 cycles of random assignment) until it reached £1057, which represents a 61% increase. During this time, mean total consumption expenditure rose from £478 to £654, which is an increase of 37%. This is in agreement with the first two stylised facts, identified above, that expenditures rise with rising income but at a slower rate. As mentioned earlier, the additional income goes partly on income tax, some is saved or invested and some is spent on non-consumption items. This includes mortgage interest payments, which appear in the EFS category ‘other expenditure’. In addition, capital repayment of mortgage is not counted as consumption expenditure.

The output from the simulation takes the form of a cross-sectional population of households and individuals within them. This is the same format as the original input file and enables a high level of flexibility in the analysis of the results. The output can be written to a micro level data-file and examined using a statistics package or represented graphically to highlight trends. The output could also be used as input for further transformations of the population. For example, an employment module could be added and expenditures copied from similar households that have the same number of working and unemployed members. A demographic module could be set up to represent the effect of population change with expenditure patterns copied from households with a similar composition.

It would not be appropriate to write out the disaggregated output dataset here. The following aggregated summary shows how the population reacts to increases in household income. This can be compared against the stylised facts mentioned above to verify that the model produces the expected results.

Table 2 shows expenditure changes for the average household. The ‘Initial Share’ is the proportion of total consumption expenditure that the average household spent on each category at the start of the simulation. The ‘Final Share’ is the budget share for the item after incomes were inflated as described above. ‘Shift in Budget Share’ is the difference between the initial and final share. Due to the stochastic nature of the simulation, there is some variation from one run to the next. As a result, it is not possible to know whether the results from a single instance are typical or unusual. Running the simulation a few times will provide information on how much random variation takes place between runs. This can be quantified in a number of ways such as calculating the difference between maximum and minimum values of the output for a particular parameter, obtained from a number of runs or from their standard deviation. It is also possible to calculate confidence intervals for the results and the next column of Table 2 gives the 95% confidence interval of the final budget share for each expenditure category. The last two columns of Table 2 show the upper and lower bounds of the confidence intervals. The results are the mean of three simulations. The variation is assumed to be normally distributed because the process generating the deviations is the random assignment scheme itself. While this gives some indication of the variability of the model, it gives no indication of the accuracy of the model specification.

Table 2

Expenditure Category	Initial Share (%)	Final Share (%)	Shift in Budget Share (%)	95% Confidence Interval (+ −) α = 0.05, n = 3	Lower CI	Upper CI
Transport	13.00	13.94	0.95	0.288	13.65	14.23
Education	1.48	2.31	0.82	0.242	2.07	2.55
Furniture	6.32	6.58	0.26	0.200	6.38	6.78
Miscellaneous	7.54	7.63	0.09	0.152	7.48	7.78
Health	1.24	1.31	0.07	0.033	1.28	1.34
Clothing	4.86	4.79	−0.08	0.095	4.70	4.89
Hotels	7.92	7.79	−0.13	0.007	7.78	7.80
Communication	2.46	2.1	−0.36	0.017	2.08	2.12
Recreation	12.26	11.89	−0.37	0.325	11.57	12.22
Alcohol	2.33	1.94	−0.39	0.039	1.90	1.98
Housing	9.94	8.92	−1.02	0.167	8.75	9.09
Food	9.81	8.14	−1.67	0.059	8.08	8.20

1. The expenditure shares to not add up to 100% because some spending goes to other items not included in these categories.

Table 2 is sorted into descending order of percentage change so that as incomes rise, households devote a greater proportion of their consumption expenditure to the goods near top of the list. These categories can be said to have a greater discretionary element than those lower down.

Table 2 indicates that the categories of ‘education’ and ‘transport’ receive a greater share of the increased expenditure than the other items. This is consistent with the ONS results mentioned above, which showed that households in the highest income decile spend more of their income in these categories than those in the lowest income decile.

At the bottom end, ‘food’ and ‘housing’, receive a diminishing share of the rising income, which is consistent with what was reported by the ONS. Finally, the budget share for ‘food’ falls as incomes rise and this is in agreement with Engel’s Law.

These results are not surprising. They are merely intended to verify that the model produces the expected results. However, the micro-level cross-sectional output can be analysed in a variety of ways and the next section shows trends in more detail and later compares the results among subsections of the population.

Figure 1 shows the results at regular points during the simulation. The share of consumption expenditure is given as a percentage, on the y-axis. Only the top two and bottom two items from Table 2 are plotted. This makes them easier to see and these are the items that exhibited the most variation. The 95% confidence intervals are shown for each data point and indicate that random variation is relatively small in this simulation.

Figure 1

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It can be seen that the lines are not monotonic and there is variation in the magnitude and direction of change from cycle to cycle. This is due to the stochastic nature of the model where the results are influenced by the characteristics of which particular donors happen to be selected in the course of each cycle. The random variations tend to cancel out over time and this is why it is preferable to complete the income change in a number of small steps rather than in one jump.

The next two sections provide further results for selected sections of the population that can be expected to have differing expenditure patterns. The first of these compares ‘older households’ and ‘families with young children’ with the average household in Section 4.4.2. After this, Section 4.4.3 gives results disaggregated by income quintile.

4.4.2 Comparison of the average household with older households and families with young children

Figure 2 shows changes in spending patterns for two types of household:

1. those with at least one person aged over 65 (older households);

2. households with one or more children under 5 (families with young children).

Figure 2

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In each case, incomes were inflated in the same way as in the previous simulation, then spending in pounds per week, for each good, was divided by the square root of the number of people in the household to compensate for household size (Buhmann et al., 1988). Next, the amount spent on each expenditure item was divided by the amount spent on that good by the average household so that the results are shown as a percentage of what the typical household spends.

The expenditure categories are shown on the x-axis: first for the older households (65+) and then for the family with young children (F). Each individual bar represents spending at each step of the simulation as incomes were increased.

These results were obtained from one simulation so there are no confidence intervals. However, the amount of variation between each cycle of random assignment can be seen in the level of irregularity between the columns within each expenditure category.

For the ‘food’ category, it is apparent that households in which at least one member is aged over 65, spend about 80% of what the average household spends. As incomes rise, this figure is largely unchanged which indicates that this group increase spending on food at about the same rate as the average household. The family with children under 5, spend more than this initially and increase spending more rapidly as incomes rise, almost reaching the same level as the average household by the end of the simulation.

The older households spend slightly over 60% of the average on ‘recreation’ at the start of the simulation. The rate of increase is much faster than the average household and reaches over 120% of average spending after incomes were inflated. By contrast, the family with young children allocate less spending than the average household to this category as incomes rise. This leads to a decline in budget share for ‘recreation’ compared with the average household. In this way, the graph gives an indication of the relative priority of allocating the extra spending due to rising incomes.

4.4.3 Analysis by household income quintile

Figure 3 shows changes in spending patterns by quintile as incomes rise, with quintile 1 being the lowest income group and quintile 5 being the highest. This is equivalised as before by dividing income and spending by the square root of household size. Results are presented for all 12 of the EFS high-level categories, as a percentage of the total spending among these categories.

Figure 3

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In Table 2 above (Section 4.4.1), ‘education’ was identified as experiencing one of the largest increases in budget share. Figure 3 shows that the increase is moderate in the lower three quintiles. Much of the change appears in quintile 4 and for quintile 5, ‘education’ forms a significant budget item. Spending on ‘transport’ increases throughout to almost double its share between quintile 1 and 5.

The budget share for ‘housing’ declines as incomes rise, with much of this taking place in the lowest three quintiles. The proportion spent on ‘food’ declines as would be expected from Engel’s Law.

This example shows how a random assignment scheme can be applied to model a set of several goods at a disaggregated level. The matching scheme used here selects a donor case and from there, any variables from the donor can be copied to the receiver. As a result, there is no restriction on the number of goods that can be modelled. In principle, every item in the EFS could be represented and the complexity of the model would rise linearly with the number of goods. Also, since the individual households are retained, the distribution of variables is preserved and output can be produced at the micro-level.

This does not mean the method is without pitfalls. As was mentioned above, selecting cases on a parameter such as increasing income means that all the variables from the case will be available for copying and some of these will be correlated with income. If the change in income really causes the change in another variable, this can be used to determine the effect of the former on the latter. However, if the change in income does not cause the change, such as when income does not cause household size to increase, this will lead to an inaccurate estimation of the effect because the simulated households change in a way that real households do not. This can be minimised by including more variables in the matching criteria to limit the household’s response to changing conditions. A sensitivity analysis can be done by adding matching criteria, one by one, until the change in output reduces to an acceptable level. A regression of the matching variables on the output variables will indicate how much of the variation is explained by the model and so give an indication of its reliability. The problem of model specification is not unique to random assignment and all modelling methods can be done well or badly whatever approach is taken.

Although random assignment preserves the individual cases, the model described above does not model trajectories. It is based on a cross-sectional dataset and assumes that households will respond to a change in circumstances by adjusting their spending patterns to become more like those that have already experienced the new conditions. This may have some plausibility but it is also reasonable to suppose that households would try to retain as much of their original behaviour as possible. It would be feasible to model this by using a longitudinal dataset and copying from donors that have experienced a similar change over time. Unfortunately, the EFS is a repeated cross-sectional survey and there are currently no large-scale longitudinal panel surveys, in the UK at least, that monitor detailed household spending categories over time.

Random assignment is related to statistical matching, which is used in microsimulation to combine data from a number of different sources or files. Here, individual cases are matched on the basis of the similarity between one or more variables which are common to both sources (Ingram et al., 2001). However, the random assignment scheme applied in this paper is essentially different from statistical matching because records are matched from within the same dataset. This makes it more akin to hot deck imputation where missing variables are obtained from a similar case that has a complete set of variables (Andridge & Little, 2010). Hot deck imputation is often used in surveys, where respondents may decline to answer some or all of the questions. This is particularly problematic because different respondents may have different propensities for omitting data so leading to biased results. A range of methods has been developed to ameliorate this problem such as propensity score matching (Rosenbaum & Rubin, 1983) and multiple imputation (Rubin, 1987). Here, the imputation process is repeated a number of times account for the variability or uncertainty in the results, which is lost in a single imputation. This allows an accurate statistical description of the resulting micro-level data file to be made. The model described above also made use of multiple runs to average out the random element of the imputation. This invites the possibility of combining the two approaches and further work in this area may prove fruitful.

Modelling household expenditure is usually done by econometric methods but there are some difficulties with this approach. These include the problem of modelling at a highly disaggregated level and limits on the number of goods that can be represented. The example model described in this paper shows that a random assignment scheme provides one way to avoid these limitations. The paper set out to illustrate the practical advantages of random assignment and the results point to areas for further research such as the possibility of matching using longitudinal data and applying multiple imputation.

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Author details

1. Tony Lawson

   Department of Sociology, University of Essex, England

   For correspondence

   tony.lawson@dunelm.org.uk