Exercise 2-1. Plotting Macroeconomic data of the US
[Link to Code]: ex2_1_PlotData_US.m
In this exercise, we will plot several macroeconomic variables for the United States and use these plots to introduce time series data and key macroeconomic indicators. Specifically, we will cover the concept of seasonality in time series data, the identity relationship between real GDP, nominal GDP, and the implicit price deflator (GDP deflator), differences between various price indices, labor market indicators and Okun’s law.
Annualization
Before diving into the details, it is important to note that the quarterly nominal and real GDP data (GDP and GDPC1) obtained from FRED are reported at annualized rates. This means that the quarterly values are multiplied by four to express what the total would be if the level of economic activity observed in that quarter were maintained for an entire year.
The reason for annualization lies in both policy practicality and comparability. Specifically, annualized figures provide the intuitive sense of “What would the annual total be if this quarterly pace continued for a full year?” This makes it easier to understand the scale of a country’s economy from an annual perspective. In fact, annualized values are commonly used in the United States, where economic reporting and policymaking often focus on annual growth trends. The use of annualized data allows analysts and decision makers to evaluate quarterly movements within a familiar yearly framework.
However, when the goal is to compare multiple quarterly macroeconomic variables, it is necessary to convert annulized GDP values to its quarterly level. To do so, the seasonally adjusted nominal and real GDP figures are divided by 4.
Also, we standardize the units of measurement by expressing all GDP values in billions of dollars. Specifically, nominal GDP figures that are not seasonally adjusted and reported in millions of dollars are converted to billions by multiplying them by 1/1,000.
Trend, Cycles, Seasonality and Transformation
Following Persons (1919), time series data are said to consist of three main unobservable components: the trend, cyclical and seasonal components. The trend refers to long-run development, while the cyclical component captures fluctuations lasting more than one year. The seasonal component represents recurring movements within a year. When applying time series analysis, it is useful to visualize and detect those properties by examining the graphical representation of the series.
Figure 1 shows the seasonally adjusted and non-seasonally adjusted series of the Nominal GDP of the United States. Specifically, one can observe that the series of nominal GDP increases in the long run, i.e. has a positive trend. There are two notable shifts in the series, corresponding to the 2008 global financial crisis and the 2020 COVID-19 shock. In addition, the non-seasonally adjusted series, depicted by the blue line, shows an apparent “jagged” pattern within each year. These are referred to as “seasonal variations”. Typical examples include sharp increases in personal consumption expenditures (PCE) during the second or fourth quarters, which coincide with the summer vacation and end-of-year holiday seasons, as well as swings in construction investment, which tends to rise during the spring and summer months and decline in the winter.
When the objective is to compare changes over time, it is desirable to eliminate seasonality prior to conducting the analysis. This is because such predictable seasonal patterns can distort the underlying relationship between quarters and make meaningful comparisons more difficult. Therefore, unless there is a specific reason, it is standard practice in macroeconomic analysis to use seasonally adjusted series, depicted in orange.
Now we delve deeper into the details and discuss various transformation methods to remove the trend and seasonality, thereby allowing us to focus on the cyclical component, which captures the underlying economic fluctuations of interest. Figure 2 above applies the quarter-on-quarter(QoQ) first-difference transformation \(( \Delta GDP_{t}=GDP_{t}-GDP_{t-1} )\) to the not-seasonally adjusted quarterly nominal GDP series of the United States. While this first-difference transformation effectively removes the trend component, seasonal variations remain evident in the data. Specifically, in most years, quarterly fluctuations occur around zero, with the nominal GDP consistently lower in the first quarter, sharply increasing in the second quarter, declining in the third quarter, and slightly increasing again in the fourth quarter. Furthermore, a structural change in seasonality can be observed over time, as the typical pattern where the fourth quarter exceeds the level of the second-quarter gradually disappears.
However, transformation methods such as Figure 2 have a limitation in that it does not account for the scale effect of the variable. Specifically, the amplitude of the series appears to increase over time, which is likely attributable to the expansion of the economy, i.e. the rising level of nominal GDP of the United States. To remove this scale effect, it is common to transform the series into quarterly growth rates. In this case, the growth rate is calculated as follows:
\[\begin{equation} qg_{gdp} = \frac{GDP_{t}-GDP_{t-1}}{GDP_{t-1}}\cdot 100 \notag \end{equation}\]The problem with this representation, however, is that it introduces asymmetry between positive and negative changes. For example, a positive change from 100 to 125 corresponds to a 25% increase, while a negative change from 125 to 100 amounts to a 20% decrease. As a result, when calculating the average growth rate for a highly volatile series, the average can be biased. In an extreme case, the average growth rate can be calculated in spite of a negative trend. To address this issue, the most commonly used method is to employ the continuous growth rate, which approximates the growth rate as follows:
\[\begin{equation} qg_{gdp}\approx (\ln{GDP_{t}}-\ln{GDP_{t-1}})\cdot 100 \notag \end{equation}\]Figure 3 above depicts the quarterly growth rate of U.S. nominal GDP, calculated as the log difference between consecutive quarters. Similar to Figure 2, the long-term trend is removed, while seasonal fluctuations remain. However, unlike the level-differenced series in Figure 2, the volatility of the series does not exhibit an increasing pattern over time. Rather, the series fluctuates around zero with a relatively stable amplitude. Interestingly, the volatility appears relatively larger prior to the 1980s, gradually decreasing thereafter, and shows a temporary spike around 2020 before stabilizing in the subsequent period.
Until now, various transformation methods that eliminate the trend while retaining seasonality in the series have been introduced. If the objective is to remove both the trend and seasonality, it is necessary to apply year-on-year (YoY) difference transformation (\(\Delta_{4}GDP_{t} = GDP_{t} - GDP_{t-4}\)) rather than the previous quarterly-on-quarter differencing. Figure 4 depicts the year-on-year differenced series of the U.S. quarterly nominal GDP. A key characteristic of this series is that both the trend and seasonal fluctuations are no longer visible. Furthermore, the annual changes are generally positive, turning negative only during periods of recession. Specifically, the series shows negative changes only during the 2008 financial crisis and the 2020 COVID-19 shock.
Similar to the QoQ differencing, the YoY differenced series in Figure 4 does not account for the scale of the variable. As a result, the amplitude of fluctuations tends to increase over time, reflecting the expansion of the nominal GDP level. To address this scale effect, it is common to transform the series into an approximate annual growth rate as follows:
\[\begin{equation} ag_{gdp} = \ln{GDP_{t}} - \ln{GDP_{t-4}}\cdot 100 \notag \end{equation}\]Figure 5 above illustrates the annual growth rate of U.S. nominal GDP. From the 1960s to 1980s, the U.S. exhibited a relatively high average nominal GDP growth rates, ranging between 2-15%, along with considerable volatility. This was largely driven by post-war industrialization, population growth, and external shocks such as the two oil crises in 1973 and 1979. Since the 1990s, however, the U.S. economy has experienced both lower growth rates and reduced volatility, commonly referred to as the Great Moderation, reflecting a combination of low inflation, slowing productivity growth, and demographic shifts. Two notable exceptions occurred in 2008 and 2020, during which volatility spiked and negative growth rates were observed. In particular, during the COVID-19 pandemic, year-on-year nominal GDP growth sharply turned negative, followed by a rapid recovery driven by expansionary fiscal spending, with growth rates reaching around 15%—levels comparable to those of the 1960s-1980s—before reverting to the recent trend levels.
Nominal and Real GDP
This section explains the definitions of nominal and real GDP, along with the implicit price index derived from them, known as the GDP deflator.
Nominal GDP is defined as the current dollar value of all final goods and services that are produced within a country within a given period of time. Formally, if an economy produces \(n\) final goods and services, and the quantity of good \(i\) produced in year \(t\) is denoted by \(y_{i,t}\) with the corresponding market price \(p_{i,t}\), then the nominal GDP in year \(t\) is the sum of prices multiplied by quantities as follows:
\[\begin{equation} NGDP_{t} = p_{1,t}\cdot y_{1,t} + p_{2,t}\cdot y_{2,t} + \cdots + p_{n,t}\cdot y_{n,t} = \sum_{i=1}^{n}{p_{i,t}\cdot y_{i,t}} \notag \end{equation}\]Defined in this way, nominal GDP changes whenever prices or quantities change. In economic analysis, our focus is maximizing utility, and utility is primarily determined by the quantity of goods and services we consume. Therefore, what ultimately matters for well-being is the quantity rather than the price level. This is why it is necessary to measure real GDP, which captures the quantity of production without being influenced by price fluctuations.
Real GDP, often referred to as constant dollar GDP, is defined by evaluating the total quantity of production at the prices of a specific base year, in order to ensure consistency across different time periods. Specifically, it multiplies the quantity of each goods and services produced each year by their corresponding prices in the chosen base year. If the base year is denoted as year \(b\) and the economy produces \(N\) goods and services, the real GDP for each year can be expressed as follows:
\[\begin{aligned} & RGDP_{b} = p_{1,b}\cdot y_{1,b} + p_{2,b}\cdot y_{2,b}+\cdots+p_{N,b}\cdot y_{N,b} \notag \\ & RGDP_{b+1} = p_{1,b}\cdot y_{1,b+1} + p_{2,b}\cdot y_{2,b+1} + \cdots + p_{N,b}\cdot y_{N,b+1} \notag \\ & \hspace{20ex}\vdots \notag \\ & \Rightarrow RGDP_{b+h} = \sum_{i=1}^{N}{p_{i,b}\cdot y_{i,b+h}}\text{ for }h=0,1,2,\cdots \notag \end{aligned}\]Based on nominal GDP and constant dollar GDP (real GDP), we can then define an implicit price index called as the GDP deflator as follows:
\[\begin{aligned} & P_{b} = \frac{NGDP_{b}}{RGDP_{b}} = \frac{p_{1,b}\cdot y_{1,b}+\cdots+p_{N,b}\cdot y_{N,b}}{p_{1,b}\cdot y_{1,b}+\cdots+p_{N,b}\cdot y_{N,b}} = 1 \notag \\ & P_{b+1} = \frac{NGDP_{b+1}}{RGDP_{b+1}} = \frac{p_{1,b+1}\cdot y_{1,b+1}+\cdots+p_{N,b+1}\cdot y_{N,b+1}}{p_{1,b}\cdot y_{1,b+1}+\cdots+p_{N,b}\cdot y_{N,b+1}} \notag \\ & \hspace{20ex}\vdots \notag \\ & \Rightarrow P_{b+h} = \frac{\sum_{i=1}^{N}{p_{i,b+h}\cdot y_{i,b+h}}}{\sum_{i=1}^{N}{p_{i,b}\cdot y_{i,b+h}}}\text{ for }h=0,1,2,\cdots \notag \end{aligned}\]From the definition of the GDP deflator, we can derive two important points. First, the price level is normalized to 1 in the base year. Second, we can establish the following identity that holds between nominal GDP, real GDP, and the price level:
\[\begin{equation} P_{t} = \frac{NGDP_{t}}{RGDP_{t}} \iff RGDP_{t} = \frac{NGDP_{t}}{P_{t}} \notag \end{equation}\]From this identity, we can derive an important implication: if the price level increases, nominal GDP must increase more rapidly than real GDP to maintain the equality.
| Variable | Quarterly | Annual |
|---|---|---|
| NGDP | 1.54 | 6.17 |
| RGDP | 0.73 | 2.92 |
| GDPDEF | 0.81 | 3.25 |
By examining the actual US macroeconomic data, one can verify if the identity empirically holds. Specifically, the average quarter-on-quarter growth rate of the nominal GDP is 1.54%, corresponding to an annualized growth rate of 6.17%. Applying the same method to real GDP, we find that the average QoQ growth rate is significantly lower, at 0.73%, equivalent to an annualized growth rate of 2.92%. According to the identity above, the gap between nominal and real GDP growth rates (approximately 3.25 percentage points) should match the average annual growth rate of the GDP deflator. Indeed, the GDP deflator’s average quarterly growth rate is 0.81%, corresponding to an annualized growth rate of 3.25%. Therefore, we can conclude that the identity holds true in actual data.
When comparing the historical data of the nominal and real GDP, a noticeable gap emerges between the two series. Below, I examine the underlying reasons for this difference. Figure 6 depicts the nominal and real GDP, expressed in natural logarithms. The key advantage of this logarithm transformation is that the vertical distance between two points can be interpreted as the approximate growth rate over the corresponding period.
The figure reveals three key takeaway points. First, the nominal GDP has increased relatively rapidly in the 1970s, but its growth rate slowed after the 1980s. Second, nominal GDP exhibits a steeper upward trend compared to the real GDP. According to the identity above, when the price level rise, the growth rate of nominal GDP must exceed that of real GDP. Indeed, as shown in Figure 7 below, both the price level and inflation regarding the GDP deflator were high during the 1970s and 1980s. Therefore, the relatively rapid increase in nominal GDP and its steeper trend compared to real GDP during this period can be attributed to this mechanism.
Third, interestingly, the level of the real GDP prior to the base year of 2017 appears to be higher than that of the nominal GDP. This is somewhat counterintuitive at first glance, considering that nominal GDP reflects real GDP adjusted for the price level. This phenomenon arises from the use of the chain-weighted method for calculating real GDP. The chain-weighted method addresses the limitation of arbitrarily fixing a single base year, especially since relative price of goods and services change continuously over time. To circumvent this issue, chain-weighted GDP calculates real GDP by first deriving real GDP between adjacent periods and then adjusting it using the geometric average of the growth rates.
Since past economies where characterized by relatively lower prices productivity, calculating real GDP using recent base-year prices tends to overestimate the level of real GDP for earlier periods. As a result, it is possible to observe a reversal in which real GDP exceeds nominal GDP prior to the base year—a phenomenon commonly seen in national statistics that apply recent base years. In this analysis, the real GDP series is calculated based on a chain-weighted method while using 2017 as the base year, which explains the observed gap between real and nominal GDP levels before and after that point.
Price Indices
The Consumer Price Index (CPI) is another widely used macroeconomic indicator for analyzing a country’s overall price level. The CPI shares a similar objective with the previously introduced GDP deflator in that both aim to measure the average price level. However, they differ fundamentally in how they are constructed. The CPI measures changes in the prices of consumed goods and services based on a “basket” of items. This basket is determined by the Bureau of Labor Statistics (BLS), which analyzes household consumption habits, and consists of various goods and services that the average household consumes within a given period. Specifically, it includes different quantities of different goods, such as 4 pounds of coffee or 12 gallons of gasoline.
Let there be \(N\) goods produced in the economy, and denote \(x_{i}\) as the fixed quantity of good \(i\) that is consumed by an average household within a given period. Then the total cost of the basket in year \(t\) can be derived by the sum of the products of the fixed quantity of each good and its corresponding market price in that period as follows:
\[\begin{equation} Cost_{t} = p_{1,t}\cdot x_{1} + p_{2,t}\cdot x_{2} + \cdots + p_{N,t}\cdot x_{N} \end{equation}\]Then the CPI in year \(t\) is defined as the ratio of the cost of the basket in that year relative to the cost of the basket in some arbitrary base year as follows:
\[\begin{aligned} P_{t}^{cpi} & = \frac{Cost_{t}}{Cost_{b}} \notag \\ & = \frac{p_{1,t}\cdot x_{1} + p_{2,t}\cdot x_{2} + \cdots + p_{N,t}\cdot x_{N}}{p_{1,b}\cdot x_{1} + p_{2,t}\cdot x_{2} + \cdots + p_{N,b}\cdot x_{N}} \notag \\ & = \frac{\sum_{i=1}^{N}{p_{i,t}\cdot x_{i}}}{\sum_{i=1}^{N}{p_{i,b}\cdot x_{i}}} \notag \end{aligned}\]A key takeaway point from the definition of the CPI is that the items included in the basket and their respective quantities are fixed. Therefore, the primary purpose of the CPI is to track how the cost of purchasing this fixed set of goods and services changes over time. If the average price level increases, the CPI will be lower than on before the base year and greater than one after the base year (although in practice, CPI data is mostly normalized to 100 for the base year).
| Variable | Quarterly | Annual | Standard Deviation |
|---|---|---|---|
| GDPDEF | 0.81 | 3.25 | 0.58 |
| CPI | 0.92 | 3.67 | 0.75 |
Figure 8-(a) shows how the log values of the GDP deflator and CPI fluctuates over time. It is clearly evident that the CPI level is consistently higher than that of the GDP deflator. Figure 8-(b) compares the quarter-on-quarter growth rates of each price index, namely the inflation rates. A key observation is that the inflation rate based on the CPI tends to be higher on average and exhibits greater volatility than that based on the GDP deflator. For example, during recession periods such as the 2008 financial crisis, the CPI inflation rate drops sharply into negative, while the GDP deflator declines only to around zero.
Specifically, Table 2 presents the quarterly and annual inflation rates and their standard deviations (as a measure of volatility) for each price index over the entire sample period. The average quarter-on-quarter inflation rate of the GDP deflator is 0.81%, which corresponds to an annualized inflation rate of 3.25%. By contrast, the CPI shows a slightly higher average QoQ inflation rate of 0.92%, equivalent to 3.67% on an annual basis. The standard deviation of CPI inflation is also higher, at 0.75, compared to 0.58 for the GDP deflator.
Such differences between the two price indices can be explained by the following two main reasons, which arise from differences in their construction and measurement purposes. First, the GDP deflator is derived based on the goods and services actually produced within a country, while the CPI is based on the goods and services consumed domestically. Consequently, the CPI not only excludes goods and services that are produced but not consumed domestically, but also includes goods produced abroad. In other words, if a certain good is produced in large quantities within the country but consumed in smaller amounts, it has a greater impact on the GDP deflator than on the CPI. For this reason, the GDP deflator is generally preferred when assessing overall price inflation for domestically produced goods and services, while the CPI is more appropriate for capturing changes in the cost of living, i.e. nominal consumption expenditures, for the average household.
Second, the two indices also differ in how they handle prices and quantities. The CPI keeps the quantities fixed at the base year level and tracks changes in prices over time, while the GDP deflator uses base year prices and reflects changes in quantities over time. As a result, the CPI tends to report higher price levels than the GDP deflator, mainly because it fixes quantities. As we learn in microeconomics, when relative prices change, consumers tend to substitute away from relatively more expensive goods toward cheaper alternatives, a behavior known as the substitution effect. However, because the CPI fixes quantities in its basket, it does not account for this substitution behavior. Therefore, in real-world situations where relative prices fluctuate over time, the CPI tends to overstate price changes, resulting in what is known as substitution bias.
Labor Market Indicators and Okun’s Law
Reference
- Gar\(\imath\)n, J., Lester, R., & Sims, E. (2018), “Intermediate macroeconomics”.
- Kirchgässner, G., Wolters, J., & Hassler, U. (2012), “Introduction to modern time series analysis,” Springer Science & Business Media.
- Persons, W. M. (1919). “Indices of business conditions: an index of general business conditions,” Harvard University Press.