Does the Consumer Price Index Accurately Measure Changes in the Living Cost?

The consumer price index (CPI) in Canada declined by .66 %, in April this year compared to the previous month. On a year-over-year basis (i.e., compared to April 2019), it declined by .15 %. This is the first year-over-year decline in the CPI observed in the month of April, since 1992. The average year-over-year inflation rate (i.e., percentage change in the CPI) for the month of April is 1.74 %. I am wondering if the CPI or the inflation rate of April 2020 really makes sense as a measure for the living cost during the lockdown of the Canadian economy.

Basically, the CPI is a weighted average of the retail prices of the goods and services consumed by households. These goods and services are classified into eight product groups. The table below shows the averages and the values in April 2020 of the year-over-year percentage change in each of these eight product groups' CPI.

Table: Year-over-Year Percentage Change in CPI, Canada, 1992:M2-2020:M4.

Product group	Average	April 2020
Food	2.24 %	3.49 %
Shelter	1.87 %	1.32 %
Household operations, furnishings and equipment	1.26 %	.24 %
Clothing and footwear	.06 %	-4.40 %
Transportation	2.44 %	-4.39 %
Health and personal care	1.45 %	1.42 %
Recreation, education and reading	1.44 %	-.26 %
Alcoholic beverages, tobacco …	2.93 %	.41 %
All items	1.80 %	-.15 %

In April, the year-over-year percentage change in the CPI for food was 3.49 %, which is much higher than its historical average of 2.24 %. The year-over-year percentage change in the CPI for shelter was also high (1.32 %) but below its historical average of 1.87 %. These two product groups along with alcoholic beverages, tobacco products and recreational cannabis were mainly the goods deemed essential and mostly the only ones that were available to households, during the lockdown imposed by the federal and the provincial governments to stop the spread of the COVID-19.

The fact that their CPIs rose on a year-over-time basis and, at the same time, the year-over-year inflation rate fell casts a doubt on the use of the CPI for all items as measure of the cost of living. It is true that the CPI for clothing and footwear and that for transportation fell by 4.4 % on a year-over-year basis, which has dragged down the inflation rate, but these goods and services were not those people in Canada mainly purchased during the lockdown.

One also ends up at the same conclusion, looking instead at the monthly growth rates of the CPI for these product groups. In April 2020, the CPI for food grew by 1.12 % compared to March, the monthly CPI for alcoholic beverages, tobacco products and recreational cannabis grew by .12 %. Shelter was the only product group deemed essential whose CPI declined (-.34 %). One can still sustain that the .34 % decline in the CPI for shelter has caused a decline in the living cost during the lockdown, as both the year-over-year and the monthly inflation rates were suggesting. To rule out this possibility, I have computed the shares of each of the eight product groups in the monthly inflation rate. They are plotted in the pie chart below. (I have computed these shares by performing a linearly constrained optimization.)

Figure: Shares of Eight Product Groups in the Monthly Inflation Rate

It turns out that shelter is the product group that accounts for the largest share of the monthly inflation rate in Canada (27.5 %). Food accounts for 16.5 % while alcoholic beverages, tobacco products and recreational cannabis accounts for 5.7 %. Health and personal care accounts for the lowest share of the monthly inflation rate (4.7 %).

Even though shelter accounts for the largest share in the inflation rate, its contribution to the change in the living cost in April was only -.09 % (i.e., -.34 % x .275) whereas the contribution of food was .18 % (i.e., 1.12 % x .164) and that of alcoholic beverages, tobacco products and recreational cannabis was .01 (i.e., .12 % x .057). My conclusion is that the CPI is a good measure of the level of prices, but it cannot accurately measure the change in the living cost in periods of economic lockdown.

Dataset and Code

Inflation and Unemployment over the Business Cycle: Comparing evidence from Canada and the United States

In an earlier post [here], I have analyzed the joint behavior of inflation and unemployment in Canada, over periods of economic expansion and contraction. In this post, I go on with this investigation using now data from the United States (US) and comparing then the new findings to the evidence from the Canadian economy.

Figure 1, below, shows the distribution of the inflation and unemployment rates in both countries over a period of time going from March 2001 to December 2019 (226 months). The distribution of the inflation rate in Canada appears to have a fatter and longer tail. In fact, in Canada, the monthly inflation rate over the period of interest fluctuated in the interval ±.54% whereas, in the US, it fluctuated between -.12% and .38%. However, both distributions are leptokurtic (i.e., their tails are fatter than those of a normally distributed variable). On the other hand, unlike the inflation rate, the distribution of the unemployment rate in the US is wider than its distribution in Canada.

Figure 1: Empirical Distributions of the Inflation and the Unemployment Rates in Canada and the US, 2001:M3-2019:M12.

It also appears in Figure 1 that, in both Canada and the US, the distribution of the unemployment rate peaks at two different points. This is what is called a bimodal distribution. In Canada, the lowest mode ( i.e., the unemployment rate associated to the first peak) has a lower probability of occurence than the highest mode. But, in the US, it is the lowest mode that has a higher probability of occurence.

Evidence 1: The distribution of the unemployment rate in Canada and the US is bimodal.

Evidence 1 implies that it is inappropriate to model the unemploymment rate assuming that it is a normally distributed variable. A normally distributed variable is bell-shaped, which implies it peaks only at one point. It is also inappropriate to model the inflation rate making such an assumption, due to the excess kurtosis in the data. (By excess kurtosis, I mean the fat tails of their distributions.) Therefore, an alternative and better way of modeling both the inflation and the unemployment rates is to use a Markov-switching multivariate normal model. A Markov-switching model assumes different unobserved states of the economy, which have their own unconditional and transition probabilities. Then, the conditional probability of an observation depends on the realized state.

Figure 2, below, plots the mixtures of two state-dependent normal distributions fitted to the inflation and the unemployment rates in Canada. Figure 3 that follows plots the estimates for the US. In both cases, the two-state Markov-switching normal model provides a better fit to the inflation rate than to the unemployment rate.

Figure 2 : Empirical Distribution and State-Dependent Distributions of the the Inflation and Unemployment Rates, Canada, 2001:M3-2019:M12.

Figure 3 : Empirical Distribution and State-Dependent Distributions of the the Inflation and Unemployment Rates, US, 2001:M3-2019:M12.

Tables 1 and 2, below, display the expected values of the inflation and the unemployment rates over the two states of the economy (expansion and contraction) respectively in Canada and the US. In Canada, the standard deviation of the unemmployment rate is .26 during periods of expansion and .48 during periods of contraction. In the US, this standard deviation is .76 and .97, respectively during periods of expansion and contraction. On the other hand, the standard deviation of the inflation rate does not change much over the two states.

Table 1: Expected values from a Markov-Dependent Mixture of Multivariate Normal models, Canada, 2001:M3-2019:M12.

	Expansion	Recession
Inflation Rate	.16 %	6.02 %
Unemployment Rate	.12 %	7.33 %

Table 2: Expected values from a Markov-Dependent Mixture of Multivariate Normal models, US, 2001:M3-2019:M12.

	Expansion	Recession
Inflation Rate	.17 %	5.00 %
Unemployment Rate	.13 %	8.56 %

Evidence 2: The volatility of the unemployment rate in Canada and the US is lower during the periods of economic expansion than during the periods of contraction.

Evidence 3: The inflation rate in Canada and the US is tends to be higher during the periods of economic expansion and lower during the periods of contraction. On the other hand, the unemployment rate tends to be higher during the periods of contraction.

While the correlation between the inflation and the unemployment rates is negative over the two states in the US, in Canada, it is negative only during the periods of contraction.

The use of higher-order Markov-switching models (i.e., models distinguishing between more than two states of the economy) has not helped improve the estimates of the marginal probabilities of the unemployment rate.

Dataset and Code

Inflation and Unemployment over the Business Cycle

I have analyzed the joint behavior of inflation and unemployment in Canada over a period of time ranging from March 2001 to December 2019 (226 months). My interest is to find out if there is any difference in the way these two variables behave during periods of economic expansion and contraction. This is done by fitting a two-state Markov-switching multivariate Student's t-model to the data.

The inflation and the unemployment rates have turned out to be more volatile over one state than the other. Decoding the states shows that the most volatile one corresponds to the periods of economic contraction that the Canadian economy experienced : the early 2000s crisis, the 2009 recession, and the oil price collapse of 2015. The volatility of the unemployment rate almost doubles during the high volatility state.

The figure below shows the densities of the inflation and the unemployment rates, and plot the marginal distributions of the fitted model. The second panel of this figure shows that the distribution of the unemployment rate is bimodal, with its lowest peak corresponding to the periods of economic expansion. It also appears that even though the Markov-switching multivariate Student's t-model has produced an accurate estimate of the expected value of the unemployment rate during periods of expansion, it overestimated its probability of occurence. The estimates for the inflation rate match the actual probabilities.

Figure : Densities and State-Dependent Distributions of the the Inflation and Unemployment Rates, Canada, 2001:M2-2019:M12.

The table below reports the expected values from the fitted models. The expected value of the inflation rate turns out to be higher during periods of expansion. On the other hand, the unemployment rate is lower during periods of expansion and higher during periods of contraction.

Table : Expected values from a Markov-Dependent Mixture of Multivariate Student's t-models, Canada, 2001:M2-2019:M12.

	Expansion	Recession
Inflation Rate	.16 %	6.01 %
Unemployment Rate	.12 %	7.28 %

The actual correlation between the inflation and the unemployment rates is positive during periods of economic expansion (.04) and negative during periods of contraction (-.02). This means that during periods of expansion, inflation is mostly driven by such supply-side factors as technological innovations, which lower prices and raise employment (decreasing unemployment, by so doing) at the same time. On the other hand, when the economy is contracting, inflation is mostly driven by factors affecting the demand-side of the economy such as preference shocks.

Dataset and Code