Producing monthly estimates of Canadian household final consumption expenditure

Author

Philip Smith

Introduction

Statistics Canada releases quarterly estimates of household final consumption expenditure (HFCE) as part of the national accounts. HFCE is presently a little over 52% of GDP and it plays a very important role in the economy. The quarterly estimates are released two months after the reference quarter and it would be beneficial if they could be published in a more timely manner. One way to address this is to calculate estimates on a monthly basis, making use of closely related monthly time series such as retail sales, monthly GDP and the consumer price index. This note describes the methodology and results of an exercise of this kind.

Quarterly estimates of household final consumption expenditure

The quarterly estimates of HFCE are published by Statistics Canada as an aggregate in the quarterly expenditure-based GDP table, 36-10-0104-01. An additional table, 36-10-0107-01, provides a breakdown of HFCE into 54 aggregates and sub-aggregates, both at current prices and as chained Fisher volume indexes. This implies a set of implicit chained Fisher price indexes, since these can be easily obtained by dividing the estimates at current prices by the chained volume indexes. Table 36-10-0124-01 provides a more detailed breakdown of HHCE with 116 aggregates and sub-aggregates, both at current prices and at constant 2012 prices. Consistent sets of estimates are available from 1961 Q1 to the latest quarter, in both unadjusted and seasonally adjusted forms. The quarterly estimates are available at the Canada level only, although provincial breakdowns are also available on an annual basis.

Statistics Canada estimates the quarterly HFCE numbers using a variety of different statistical sources. There is no quarterly survey of consumer expenditure per se. Among the more important sources1 used by Statistics Canada are:

  • The retail trade survey
  • The consumer price index survey
  • The survey of food services and drinking places
  • The survey of international travellers entering or returning to Canada

These are by no means the only data sources used, as Statistics Canada also draws on more specific data, some from administrative sources, to improve the estimates of individual components of HFCE. In addition, the quarterly estimates are eventually benchmarked to data from the annual supply and use accounts, which become available with a lag of about three years.

Nevertheless the four sources just listed are among the most important ones for purposes of the unbenchmarked quarterly estimates. The first two, in particular, are critical because they provide information about consumption patterns in a great many of the individual HFCE components. The fortunate thing about these four sources is that they are all available on a monthly basis.2

Estimating a monthly time series given a quarterly time series

The problem of how best to estimate a higher frequency time series such as a monthly one, given a lower frequency series such as a quarterly one, has been addressed in several papers over the years. Some of these are recorded in the references list below.

A simple way to do it is to fit a high-frequency (i.e. monthly in this case) polynomial curve through the low frequency (i.e. quarterly in this case) estimates, selecting the polynomial to minimize a loss function. Since it is desirable to have the polynomial be as smooth as possible - we do not want the interpolated and projected monthly estimates to exhibit spurious blips - a good choice for the loss function might be the sum of squared monthly changes in the polynomial. Frank Denton, a former Statistics Canada employee, wrote a paper on this approach(Denton 1971) as did Estelle Dagum and Pierre Cholette(Dagum and Cholette 2006), also former Statistics Canada employees.

A better way forward is to make use of related monthly series when these are available. Gregory Chow and A-L Lin suggested such a method.(Chow and Lin 1971) Simply postulate a linear relationship between the unknown target monthly time series ym and the known monthly indicator or indicators Xm. This postulated relationship, in the form of an equation, can be time aggregated to get the corresponding quarterly relationship, the parameters of which can be estimated in a quarterly regression using generalized least squares. These parameter estimates can then be used in the monthly version of the linear relationship for ym to generate monthly time series estimates. In the present application, the Chow-Lin approach to temporal disaggregation is employed.

The details of the calculations

To put the above methodology into effect, the first step is to collapse some of the detail provided in Statistics Canada’s quarterly HFCE breakdown:

  • The separate categories ‘food’ and ‘non-alcoholic beverages’ became one aggregate category.

  • The three ‘health’ sub-components, which are ‘medical products, appliances and equipment’, ‘out-patient services’ and ‘hospital services’, were combined as one aggregate ‘health’ component.

  • The ‘insurance’, ‘financial services indirectly measured’ and ‘other financial services’ components were combined as ‘insurance and financial services’.

  • The ‘personal care’, ‘personal effects’, ‘social services’ and ‘other services’ components were combined into ‘miscellaneous goods and services’.

It would be difficult to find suitable related indicators for these lower-level consumption categories, so aggregating them simplifies the task somewhat. This leaves 33 HFCE components for which related indicators are needed. Higher-level aggregates can also be calculated from the 33 HFCE components just mentioned, bringing the total number of target series to 42.

There is one other very useful related indicator that has not been discussed so far: monthly real gross domestic product by industry. This indicator measures production rather than consumption, so it is not very useful for many of the goods consumption components where inventory fluctuations, imported supplies, and trade and transportation margins complicate the picture. But for the services consumption components, industry value-added can often be a very useful indicator. Indeed, production and consumption are close to being the same thing for many service products.

A difficulty with using the monthly real GDP indicators though is that they measure volume, whereas the first goal is to estimate HFCE at current prices. This difficulty can be readily addressed in some cases by including the corresponding CPI as a second indicator. Thus the monthly movement of the HFCE component can, in some cases, be modelled as a combination of distinct volume and price effects.

An example

A simple example is shown below to illustrate the process. The example is for quarterly household expenditure on footwear, at current prices, interpolated and extrapolated using monthly retail sales by shoe stores, also at current prices, as a related indicator. The red line is the former, extracted from national accounts table 36-10-0107-01, and the blue line shows the monthly estimates of household expenditure on footwear, obtained using the Chow-Lin method with the related indicator monthly retail sales by shoe stores, retrieved from Statistics Canada table 20-10-0008-01.

A log linear model with a first-order autoregressive error assumption was assumed. The intercept and slope parameters from the regression with quarterly data have t-values of 4.85 and 23.23 respectively, indicating a statistically significant relationship as expected. The autoregressive error parameter is estimated at 0.968 and the adjusted R-squared is 0.948.

Chart 1 shows the full sample period from 2010 to 2022. During more normal times, from 2010 to 2019, the monthly variations are small while the fluctuations have been much wider during and after the pandemic disruption.

The method, in this case, constrains the estimated monthly series to be the same as the quarterly series when temporally aggregated by averaging. Since the quarterly series (in red) is plotted at the first month of the quarter, the estimated monthly series (in blue) is plotted at that same month and the following two. In this instance, the monthly series points to a substantially deeper decline and quicker recovery during the pandemic months between March and June than is evident in the quarterly series.

Of course, not all product groups are as easy to deal with as footwear, which happens to have its own exclusive category of retailers. It can be substantially more difficult to achieve a good fit between the quarterly series for some product groups and the corresponding indicator or indicators used to interpolate and extrapolate it. There are a wide variety of situations.

Statistical results

Table 1 summarizes the results for the statistical models linking the quarterly components of HFCE at current prices to their corresponding monthly indicators. As noted, they are log-linear models with an autoregressive error term.

Table 1
Consumer expenditure at current prices: model results
Estimation period 2010 Q1 to 2022 Q2, seasonally adjusted1
ARSQ SIGMA RHO MEAN COV X1 X2
Food and non-alcoholic beverages 0.669 685 0.998 106,997 0.640 RTS —
Alcoholic beverages 0.990 165 0.686 22,167 0.744 RTS —
Tobacco 0.981 99 0.815 16,694 0.596 CPI —
Clothing 0.981 350 0.916 38,331 0.912 RTS —
Footwear 0.945 103 0.968 7,054 1.463 RTS —
Paid rental fees for housing 0.713 246 0.999 59,829 0.411 CPI —
Imputed rental fees for housing 0.913 453 0.993 178,168 0.254 GDP —
Furniture, furnishings, carpets and other floor coverings 0.481 358 0.997 18,610 1.922 RTS —
Household appliances 0.326 186 0.998 8,885 2.096 RTS —
Other goods and services related to the dwelling and property 0.843 849 0.565 26,227 3.239 CPI —
Purchase of vehicles 0.915 1,241 0.947 71,762 1.729 RTS —
Communications 0.268 208 0.999 30,394 0.684 CPI —
Other recreational items and equipment, garden products and pets 0.231 502 0.998 24,162 2.079 RTS —
Food and beverage services 0.991 752 0.595 65,500 1.148 FSDP —
Expenditure by Canadians abroad 0.906 1,530 0.983 38,926 3.930 ITERC —
Expenditure by non-residents in Canada 0.872 738 0.994 −28,165 −2.619 ITERC —
Maintenance and repair of the dwelling 0.962 55 0.756 3,573 1.544 CPI RTS
Water supply and sanitation services 0.310 62 0.999 7,678 0.812 CPI GDP
Electricity, gas and other fuels 0.222 571 0.972 28,603 1.997 RTS CPI
Household textiles 0.732 83 0.923 3,524 2.346 RTS GDP
Tools and equipment for house and garden 0.956 405 0.194 6,111 6.625 RTS GDP
Health 0.935 425 0.997 48,623 0.873 CPI GDP
Operation of transport equipment 0.835 1,616 0.952 75,370 2.144 CPI GDP
Transport services 0.903 662 0.990 20,517 3.225 CPI GDP
Audio-visual, photographic and information processing equipment 0.103 214 0.977 14,486 1.475 CPI WTS
Other major durables for recreation and culture 0.621 287 0.923 8,413 3.416 RTS CPI
Recreational and cultural services 0.906 944 0.703 40,111 2.354 CPI GDP
Newspapers, books and stationery −0.042 188 0.572 6,973 2.698 CPI GDP
Education 0.364 189 0.998 18,929 0.996 CPI GDP
Accommodation services 0.933 231 0.979 10,725 2.151 CPI GDP
Insurance and financial services 0.921 865 0.902 102,404 0.845 CPI GDP
Miscellaneous goods and services 0.822 1,884 0.704 59,248 3.179 RTS CPI
@PhilSmith26
1 ARSQ is the adjusted R-squared. SIGMA is the estimated standard deviation of the error term. RHO is the estimated autoregressive error parameter. MEAN is the average value of the dependent variable over the sample period. COV is the coefficient of variation, equal to SIGMA divided by MEAN times 100. X1 and X2 is/are the source(s) for the explanatory variable(s). RTS stands for retail trade survey, CPI for consumer price index survey, GDP for monthly real gross domestic product estimates, FSDP for the food services and drinking places survey, and ITERC for the international travellers entering or returning to Canada survey. Estimates derived using time disaggregation methods credited to G.C. Chow and A.-L. Lin, and data from Statistics Canada tables 36-10-0107-01, 20-10-0008-01, 18-10-0004-01, 36-10-0434-01, 21-10-0019-01 and 24-10-0054-01. 2022-09-29 08:43:52

One of the more difficult series to model well is expenditure by Canadians abroad. There is no right-on-target monthly indicator series. The one used in the model reported in the table is the monthly number of Canadians returning home from another country during the month. Travel expenditures abroad accounted for 1.8% of total consumer expenditures in the second quarter of 2022.

Another difficult one is expenditure on tools and equipment for house and garden. The monthly indicators used in this case are retail sales of building material and garden equipment and supplies dealers, and real GDP of hardware manufacturers. The coefficient of variation in this case is 6.5% but fortunately it accounted for only 0.7% of total consumer expenditure in 2022 Q2.

There is of course some scope to improve these models and this is on the future agenda.

Monthly estimates at constant prices

The model and results discussed so far pertain to household expenditure at current prices. But estimates at constant prices can also be developed using a similar approach. The first step is to retrieve the national accounts estimates for household expenditure at constant (2012) prices from table 36-10-0124-01 and divide each of these time series into the corresponding time series estimates at current prices, thereby obtaining the quarterly national accounts implicit price indexes (IPIs) for each of the 42 expenditure categories. The second step is to find appropriate related indicators for these IPIs. Fortunately excellent ones are available in the detailed product categories of the consumer price index. Statistics Canada itself uses the CPI as a deflator for many, although not all of the product groups it deflates, so these indicators should work extremely well in many if not all cases. However, since Statistics Canada develops its current and constant price estimates with a greater level of product detail than the 42 categories considered here, these models will vary in their accuracy. The final step is to use the interpolated and extrapolated IPIs to deflate the monthly estimates at current prices, to obtain monthly estimates at constant (2012) prices.3

The models for the monthly price indexes are all based on consumer price indexes. They are summarized in Table 2 below.

Table 2
Consumer expenditure price indexes: model results
Estimation period 2010 Q1 to 2022 Q2, seasonally adjusted1
ARSQ SIGMA RHO MEAN COV
Food and non-alcoholic beverages 0.982 0.789 0.582 107.0 0.737
Alcoholic beverages 0.992 0.248 0.727 106.2 0.233
Tobacco 0.992 0.740 0.818 125.0 0.592
Clothing 0.854 0.555 0.494 101.2 0.548
Footwear 0.131 1.095 0.627 100.8 1.087
Paid rental fees for housing 0.999 0.232 0.000 105.3 0.220
Imputed rental fees for housing 0.464 0.249 0.998 105.3 0.236
Maintenance and repair of the dwelling 0.892 0.292 0.986 107.6 0.271
Water supply and sanitation services 0.996 1.148 0.390 117.9 0.974
Electricity, gas and other fuels 0.869 0.653 0.925 106.5 0.613
Furniture, furnishings, carpets and other floor coverings 0.723 0.704 0.942 103.7 0.679
Household textiles 0.986 0.749 0.447 104.6 0.716
Household appliances 0.746 0.368 0.984 102.4 0.360
Tools and equipment for house and garden 0.481 0.498 0.975 104.0 0.479
Other goods and services related to the dwelling and property 0.317 0.354 0.998 105.3 0.336
Health 0.951 0.505 0.732 104.6 0.483
Purchase of vehicles 0.941 1.594 0.618 108.8 1.465
Operation of transport equipment 0.866 1.638 0.792 99.2 1.651
Transport services 0.140 0.916 0.987 104.5 0.876
Communications 0.749 0.337 0.989 103.0 0.327
Audio-visual, photographic and information processing equipment 0.615 0.534 0.998 88.4 0.604
Other major durables for recreation and culture 0.526 0.832 0.983 110.8 0.751
Other recreational items and equipment, garden products and pets 0.880 0.587 0.766 103.0 0.571
Recreational and cultural services 0.989 0.603 0.650 108.3 0.556
Newspapers, books and stationery 0.342 0.505 0.999 111.3 0.453
Education 0.987 0.770 0.623 110.8 0.695
Food and beverage services 0.997 0.126 0.887 109.5 0.115
Accommodation services 0.056 2.122 0.974 106.4 1.993
Insurance and financial services 0.027 0.600 0.805 101.2 0.593
Miscellaneous goods and services 0.472 0.372 0.994 105.4 0.353
Expenditure by Canadians abroad 0.200 1.650 0.966 116.1 1.422
Expenditure by non-residents in Canada 0.104 1.215 0.998 110.3 1.101
@PhilSmith26
1 ARSQ is the adjusted R-squared. SIGMA is the estimated standard deviation of the error term. RHO is the estimated autoregressive error parameter. MEAN is the average value of the dependent variable over the sample period. COV is the coefficient of variation, equal to SIGMA divided by MEAN times 100. The model in each case includes a constant term and a CPI component. Estimates derived using time disaggregation methods credited to G.C. Chow and A.-L. Lin, and data from Statistics Canada tables 36-10-0107-01, 20-10-0008-01, 18-10-0004-01, 36-10-0434-01, 21-10-0019-01 and 24-10-0054-01. 2022-09-29 08:43:53

The statistical results are very good, with most coefficients of variation well below 1%. Accommodation services has the largest COV at 2.0% while purchase of vehicles, operation of transport equipment, expenditures by Canadians abroad and expenditure by non-residents in Canada might benefit from further work.

Continuing the footwear example

Using the interpolated monthly price indexes to deflate the monthly consumer expenditure at current prices estimates yields monthly consumer expenditure at constant 2012 prices. The chart below shows the results for the footwear example discussed earlier.

Software

Proceeding as described above to estimate monthly HFCE time series using related series was done using the R programming language in the RStudio environment, along with the ‘tempdisagg’ package. The latter is a set of R functions implemented by Christoph Sax and Peter Steiner(Sax and Steiner 2013). It is a very flexible package with a wide range of options for different methods of temporal disaggregation.

The source code for this project is freely available on GitHub.

RStudio’s recently implemented and highly recommended quarto package was used to write and publish this documentation to the Internet.

Conclusions

The project has successfully created monthly HFCE estimates from 2010 to 2022, in value, price and volume terms. The estimates are consistent with Statistics Canada’s published quarterly results in the senses that (1) the official estimate at current prices for any one of the 42 HFCE series, in any quarter, seasonally adjusted at annual rates, is equal the average of the three corresponding monthly values in that quarter and (2) the official estimate of the price index (2012=100) for any one of the 42 HFCE series, in any quarter, seasonally adjusted, is equal the average of the three corresponding monthly values in that quarter. The monthly estimates at constant 2012 prices are also approximately, though not exactly equal to Statistics Canada corresponding published quarterly estimates when averaged over any quarter. The relationship is approximate because they are derived indirectly, using the interpolated price indexes for deflation.4

The plan is to use this model on a monthly basis, reporting summary results on Twitter and continuing efforts to improve the estimates. These initial results provide 12-and-a-half-year time series for 42 aggregates and sub-aggregates of household expenditure on goods and services which may be useful in other studies. They also provide, and will continue to provide forward monthly projections that may shed light on current economic developments as economy-watchers await the arrival of the quarterly national accounts two months after the quarter has ended.

References

Chow, G. C., and A.-L. Lin. 1971. “Best Linear Unbiased Interpolation, Distribution, and Extrapolation of Time Series by Related Series.” The Review of Economics and Statistics, 53 (4): 372–75.
Dagum, Estelle B., and P. A. Cholette. 2006. Benchmarking, Temporal Distribution, and Reconciliation Methods for Time Series. Springer-Verlag, New York.
Denton, F. T. 1971. “Adjustment of Monthly or Quarterly Series to Annual Totals: An Approach Based on Quadratic Minimization.” Journal of the American Statistical Association, 66: 99–102.
Sax, Christoph, and Peter Steiner. 2013. “Temporal Disaggregation of Time Series.” The R Journal 5 (2): 1–87.

Footnotes

  1. Imports of specific products could be another useful set of related indicators for some consumption goods, although they have not been tried at this stage of the project.↩︎

  2. While retail sales are a key source for estimating consumer expenditure, it should be remembered that not all such sales are to consumers and not all sales to consumers are done through retail channels.↩︎

  3. The estimates at constant (2012) prices are the target of this exercise rather than the chained Fisher volume estimates because the latter are not additive, which makes them much more difficult to work with.↩︎

  4. The unweighted average of the monthly interpolated price indexes within a quarter is constrained to equal Statistics Canada’s published quarterly price index, when in fact the average should be a weighted one. This difference is minor and does not materially affect the monthly estimates at constant 2012 prices.↩︎